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 | C2×#33982 | 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 | |
975 conjugacy classes of subgroups; only the 323 normal subgroups are shown.
This group has 975 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:
(3,17)(4,23)(5,8)(6,25)(7,33)(10,39)(12,46)(13,16)(14,48)(15,55)(18,59)(19,67)(20,69)(21,72)(22,64)(24,78)(26,60)(27,86)(28,74)(29,77)(30,83)(31,37)(32,62)(34,70)(35,84)(36,63)(38,61)(40,99)(41,101)(42,103)(43,104)(44,105)(45,53)(47,91)(49,96)(50,115)(51,107)(52,113)(54,79)(56,102)(57,98)(58,97)(65,112)(66,111)(68,87)(71,82)(73,81)(75,76)(80,90)(85,122)(88,125)(89,93)(92,124)(94,121)(95,123)(100,116)(106,128)(108,109)(110,118)(114,127)(117,120)(119,126);(1,2)(3,18)(4,12)(5,13)(6,14)(7,15)(8,16)(9,11)(10,22)(17,59)(19,58)(20,65)(21,66)(23,46)(24,79)(25,48)(26,49)(27,50)(28,51)(29,43)(30,52)(31,53)(32,91)(33,55)(34,56)(35,44)(36,94)(37,45)(38,40)(39,64)(41,71)(42,73)(47,62)(54,78)(57,95)(60,96)(61,99)(63,121)(67,97)(68,119)(69,112)(70,102)(72,111)(74,107)(75,108)(76,109)(77,104)(80,117)(81,103)(82,101)(83,113)(84,105)(85,114)(86,115)(87,126)(88,106)(89,110)(90,120)(92,100)(93,118)(98,123)(116,124)(122,127)(125,128);(1,3,5,19,9,22,8,38)(2,10,13,40,11,18,16,58)(4,24,26,80,29,62,28,89)(6,32,30,90,35,78,34,93)(7,36,31,73,37,95,33,21)(12,47,49,110,43,79,51,117)(14,54,52,118,44,91,56,120)(15,57,53,66,45,94,55,42)(17,60,61,23,64,74,67,77)(20,70,68,84,71,83,92,25)(27,87,85,69,88,124,75,82)(39,96,97,46,59,107,99,104)(41,102,100,105,65,113,119,48)(50,116,114,101,106,126,108,112)(63,122,72,86,123,76,81,125)(98,127,103,115,121,109,111,128);(2,11)(10,18)(12,43)(13,16)(14,44)(15,45)(39,59)(40,58)(41,65)(42,66)(46,104)(47,79)(48,105)(49,51)(50,106)(52,56)(53,55)(54,91)(57,94)(96,107)(97,99)(98,121)(100,119)(101,112)(102,113)(103,111)(108,114)(109,127)(110,117)(115,128)(116,126)(118,120)Polycyclic presentation:
2^10|c^2=f,e^2=gh,f^2=j,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^a=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, #72322901, #72322960 |
| maximal normal subgroup of | #72321422, #72322901, #72322960 |
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, 3, F.6);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetPower(coll, 6, 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, 1, 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);