Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#1415 |
| commutator | C32 | C23×C4 |
| Frattini, cosocle | C2×C32 | C24 |
| prime radical | #7625194 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#131 ◁ C2×#504 ◁ C2×#2258 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
962 conjugacy classes of subgroups; only the 323 normal subgroups are shown.
This group has 962 conjugacy classes of subgroups — too many to draw. Only its 323 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(3,15,17,58)(4,21)(5,27)(6,29)(7,33)(9,36,16,55)(11,44)(12,49)(13,51)(14,43)(18,66,64,75)(19,68,65,74)(20,69,35,61)(22,28,59,57)(23,31)(24,79)(25,83)(26,56)(30,78)(32,60,93,123)(34,90)(37,97,62,109)(38,99,63,108)(39,100,54,96)(40,94)(41,46)(42,103)(45,50,73,87)(47,113)(48,115)(52,112)(53,95,92,121)(67,80,88,125)(70,71)(72,81)(76,91,84,89)(77,122)(82,124)(85,86)(98,114,118,126)(101,128)(102,107)(104,117)(105,106)(110,120,116,119)(111,127);(1,2)(3,16)(4,11)(5,12)(6,13)(7,14)(8,10)(9,17)(15,55)(18,62)(19,63)(20,54)(21,44)(22,73)(23,46)(24,47)(25,48)(26,40)(27,49)(28,87)(29,51)(30,52)(31,41)(32,92)(33,43)(34,42)(35,39)(36,58)(37,64)(38,65)(45,59)(50,57)(53,93)(56,94)(60,121)(61,100)(66,109)(67,118)(68,108)(69,96)(70,105)(71,106)(72,107)(74,99)(75,97)(76,116)(77,111)(78,112)(79,113)(80,126)(81,102)(82,101)(83,115)(84,110)(85,104)(86,117)(88,98)(89,120)(90,103)(91,119)(95,123)(114,125)(122,127)(124,128);(1,3)(2,9)(4,22)(5,28)(6,32)(7,35)(8,17)(10,16)(11,45)(12,50)(13,53)(14,54)(15,56)(18,27)(19,29)(20,33)(21,58)(23,64)(24,80)(25,84)(26,59)(30,91)(31,57)(34,93)(36,94)(37,49)(38,51)(39,43)(40,73)(41,87)(42,92)(44,55)(46,62)(47,114)(48,116)(52,120)(60,124)(61,72)(63,103)(65,90)(66,122)(67,78)(68,77)(69,83)(70,75)(71,74)(76,81)(79,123)(82,125)(85,89)(86,88)(95,128)(96,107)(97,127)(98,112)(99,111)(100,115)(101,126)(102,110)(104,119)(105,109)(106,108)(113,121)(117,118);(2,10)(9,16)(11,40)(12,41)(13,42)(14,43)(36,55)(37,62)(38,63)(39,54)(44,94)(45,73)(46,49)(47,101)(48,102)(50,87)(51,103)(52,104)(53,92)(95,121)(96,100)(97,109)(98,118)(99,108)(105,127)(106,111)(107,115)(110,116)(112,117)(113,128)(114,126)(119,120)Polycyclic presentation:
2^10|a^2=f,e^2=ghj,g^2=hi,h^2=ij,i^2=j,c^a=ce,c^b=cj,d^b=dj,e^a=eg,e^c=egj,f^c=fj,g^a=gh,g^c=gh,h^a=hi,h^c=hi,i^a=ij,i^c=ij160 conjugacy classes — view the full character table.
| maximal subgroup of | #72321422, #72322916, #72322972 |
| maximal normal subgroup of | #72321422, #72322916, #72322972 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);