Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152629 |
| commutator | C2×C32 | C22×C4 |
| Frattini, cosocle | C22×C32 | C23 |
| prime radical | #512941 | C2 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ C2×#2258 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 3 generators on 128 points:
(1,2,5,11,31,28,78,84,7,13,32,48,53,41,98,99,39,51,92,109,110,97,127,126,94,86,108,88,37,26,80,83,9,12,34,47,91,87,124,125,40,50,93,90,36,29,79,85,8,14,33,49,52,38,95,89,35,27,46,30,6,4,23,25)(3,15,18,42,64,61,113,117,20,55,70,101,107,77,121,116,75,45,68,105,112,96,128,119,111,81,100,63,54,24,66,58,22,10,17,43,65,82,120,115,76,44,67,104,73,62,114,118,21,56,71,102,106,74,123,122,72,60,103,69,19,16,57,59);(1,3)(2,10)(4,24)(5,18)(6,19)(7,20)(8,21)(9,22)(11,43)(12,15)(13,44)(14,45)(16,26)(17,34)(23,57)(25,58)(27,81)(28,82)(29,77)(30,63)(31,64)(32,70)(33,71)(35,72)(36,73)(37,54)(38,96)(39,75)(40,76)(41,62)(42,47)(46,103)(48,104)(49,105)(50,55)(51,56)(52,106)(53,107)(59,83)(60,86)(61,87)(65,91)(66,80)(67,93)(68,92)(69,88)(74,97)(78,113)(79,114)(84,115)(85,116)(89,119)(90,101)(94,111)(95,123)(98,121)(99,118)(100,108)(102,109)(110,112)(117,125)(120,124)(122,126)(127,128);(1,4)(2,6)(3,16,22,24)(5,30)(7,38)(8,41)(9,26)(10,19,15,54)(11,46)(12,37)(13,52)(14,53)(17,63,18,69)(20,74,76,96)(21,77,75,62)(23,25)(27,31)(28,35)(29,39)(32,49)(33,48)(34,88)(36,51)(40,97)(42,100,43,103)(44,106,55,112)(45,107,56,73)(47,108)(50,110)(57,59,66,58)(60,65,81,64)(61,111,82,72)(67,105,70,102)(68,104,71,101)(78,89)(79,99)(80,83)(84,95)(85,98)(86,91)(87,94)(90,92)(93,109)(113,122,120,119)(114,116,121,118)(115,123,117,128)(124,126)(125,127)Polycyclic presentation:
2^10|a^2=e,c^2=deg,d^2=gh,e^2=gh,g^2=hi,h^2=ij,i^2=j,b^a=bj,c^a=cd,c^b=cf,d^b=dj,d^c=dg,e^c=eg,f^a=fj,f^c=fj,g^c=gh,h^c=hi,i^c=ij148 conjugacy classes — view the full character table.
| maximal subgroup of | #72322773, #72322933, #72322947 |
| maximal normal subgroup of | #72322773, #72322933, #72322947 |
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.5);;
gap> SetPower(coll, 3, F.4*F.5*F.7);;
gap> SetPower(coll, 4, F.7*F.8);;
gap> SetPower(coll, 5, F.7*F.8);;
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, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);