Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13041 |
| commutator | #124 | C23×C4 |
| Frattini, cosocle | C2×#124 | C24 |
| prime radical | C2×#33558 | C2 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,9)(4,18)(5,24)(6,27)(7,11)(8,13)(10,22)(12,30)(14,41)(15,32)(16,33)(17,35)(19,50)(20,48)(21,54)(23,49)(25,34)(26,39)(28,60)(29,57)(31,45)(36,58)(37,53)(38,55)(40,59)(42,56)(43,64)(44,63)(46,62)(47,52)(51,61);(1,3)(2,9)(4,19,23,52)(5,25,7,15)(6,28)(8,17)(10,36,38,64)(11,34,24,32)(12,40)(13,35)(14,42,44,53)(16,29,46,26)(18,47,49,50)(20,51,21,45)(22,43,55,58)(27,60)(30,59)(31,54,61,48)(33,39,62,57)(37,63,56,41);(1,4,5,21,8,20,7,23)(2,10,11,38,13,37,24,56)(3,14,15,44,17,43,25,58)(6,22,26,42,30,53,29,55)(9,31,32,50,35,47,34,61)(12,18,39,49,27,48,57,54)(16,45,28,19,46,52,59,51)(33,41,60,63,62,64,40,36);(4,20)(10,37)(14,43)(18,48)(19,51)(21,23)(22,53)(31,47)(36,63)(38,56)(41,64)(42,55)(44,58)(45,52)(49,54)(50,61)Polycyclic presentation:
2^10|b^2=g,c^2=e,e^2=j,f^2=ij,h^2=j,b^a=bg,c^a=cf,c^b=cfhij,d^c=dj,e^a=eij,e^b=ei,f^a=fij,f^b=fh,g^c=ghj,g^e=gj,g^f=gi,h^a=hi,h^b=hij,h^g=hj,i^b=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72286117, #72286528, #72286670 |
| maximal normal subgroup of | #72286117, #72286528, #72286670 |
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, 2, F.7);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);