Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #364471 |
| commutator | C23×C4 | C23×C4 |
| Frattini, cosocle | C23×D4 | C24 |
| prime radical | #6437636 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C2×#3485 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
5065 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)(3,13,16,36)(4,9)(5,21,22,10)(6,12)(7,11)(8,17,27,43)(14,34,38,57)(15,40)(18,31,44,28)(19,30)(20,29)(23,48,49,32)(24,33)(25,39,51,61)(26,41)(35,42)(37,58)(45,55,63,53)(46,54)(47,62)(50,59)(52,64)(56,60);(1,3)(2,8)(4,14)(5,15)(6,17,24,43)(7,16)(9,25)(10,26)(11,27)(12,13,33,36)(18,37)(19,39,46,61)(20,38)(21,47)(22,42)(23,41,49,62)(28,50)(29,51)(30,34,54,57)(31,52)(32,35,48,40)(44,60)(45,59,63,64)(53,56,55,58);(1,4,7,20)(2,9,11,29)(3,14,16,38)(5,18,22,44)(6,19,24,46)(8,25,27,51)(10,31,21,28)(12,30,33,54)(13,34,36,57)(15,37,42,60)(17,39,43,61)(23,45,49,63)(26,52,47,50)(32,55,48,53)(35,58,40,56)(41,59,62,64);(1,5)(2,10)(3,15)(4,18)(6,23)(7,22)(8,26)(9,28)(11,21)(12,32)(13,35)(14,37)(16,42)(17,41)(19,45)(20,44)(24,49)(25,50)(27,47)(29,31)(30,53)(33,48)(34,56)(36,40)(38,60)(39,59)(43,62)(46,63)(51,52)(54,55)(57,58)(61,64)Polycyclic presentation:
2^10|a^2=e,b^2=h,c^2=j,g^2=j,b^a=bg,c^a=ceij,d^a=df,d^c=df,e^b=ej,e^d=ej,f^a=fj,g^b=gi,h^a=hij,h^g=hj,i^b=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72303157, #72303302, #72303440 |
| maximal normal subgroup of | #72303157, #72303302, #72303440 |
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, 2, F.8);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, 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);