Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #21405 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C23×C8 | C24 |
| socle | C2 | #363162 |
| prime radical | #10575331 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ C22×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
4161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(10,36)(20,52)(31,62)(37,63)(38,60)(39,54)(43,64)(53,55);(1,2)(3,9)(4,20)(5,12)(6,11)(7,30)(8,13)(10,22)(14,26)(15,43)(16,41)(17,50)(18,51)(19,34)(21,55)(23,52)(24,54)(25,60)(27,42)(28,32)(29,40)(31,44)(33,49)(35,48)(36,58)(37,61)(38,57)(39,59)(45,64)(46,63)(47,53)(56,62);(1,3,8,19)(2,9,13,34)(4,21)(5,16,27,49)(6,28,29,17)(7,18,26,48)(10,37)(11,32,40,50)(12,41,42,33)(14,35,30,51)(15,25)(20,53)(22,46)(23,47)(24,44)(31,39)(36,63)(38,43)(45,57)(52,55)(54,62)(56,59)(58,61)(60,64);(1,4,6,24,7,25,27,58,8,23,29,59,26,57,5,22)(2,10,12,38,14,39,40,52,13,36,42,60,30,54,11,20)(3,15,17,46,18,47,16,44,19,45,28,61,48,21,49,56)(9,31,33,53,35,63,32,43,34,62,41,55,51,37,50,64)Polycyclic presentation:
2^10|c^2=fj,d^2=g,e^2=ij,g^2=i,i^2=j,b^a=bf,c^b=cf,d^a=dh,d^b=de,d^c=dij,e^b=ei,f^d=fj,g^b=gij,g^c=gj,h^b=hj,i^b=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72307810, #72308342 |
| maximal normal subgroup of | #72307810, #72308342 |
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*F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);