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, socle | C2 | #366548 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#33531 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ C2×#1757 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
3440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,9)(3,14)(5,26)(6,29)(8,28)(10,37)(11,34)(12,30)(13,43)(15,33)(16,48)(17,18)(19,31)(20,51)(21,45)(22,54)(23,57)(24,56)(27,32)(35,36)(39,47)(42,64)(46,60)(49,59)(50,58)(52,61)(53,55)(62,63);(1,2,5,9,28,35,29,43,7,12,6,30,8,13,26,36)(3,15,17,44,32,33,27,37,18,39,14,40,19,47,31,10)(4,20,22,51,56,42,57,61,25,11,23,34,24,52,54,64)(16,49,48,62,53,60,58,38,21,46,45,63,50,59,55,41);(1,3)(2,10)(4,21)(5,27)(6,31)(7,18)(8,32)(9,33)(11,41)(12,37)(13,44)(14,26)(15,43)(16,25)(17,29)(19,28)(20,38)(22,55)(23,58)(24,50)(30,47)(34,60)(35,40)(36,39)(42,62)(45,57)(46,61)(48,54)(49,64)(51,59)(52,63)(53,56);(1,4)(2,11)(3,16)(5,22)(6,23)(7,25)(8,24)(9,34)(10,38)(12,20)(13,42)(14,45)(15,46)(17,48)(18,21)(19,50)(26,54)(27,58)(28,56)(29,57)(30,51)(31,55)(32,53)(33,59)(35,52)(36,61)(37,41)(39,49)(40,62)(43,64)(44,63)(47,60)Polycyclic presentation:
2^10|b^2=e,e^2=ij,f^2=ij,i^2=j,b^a=bf,c^a=cfh,c^b=cfj,d^a=dg,d^b=dj,d^c=dj,e^a=ei,e^c=eij,f^a=fi,f^b=fj,f^c=fi,g^c=gj,h^d=hj,i^a=ij,i^c=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72320921, #72343925, #72343980 |
| maximal normal subgroup of | #72320921, #72343925, #72343980 |
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.5);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
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);