Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #6453969 |
| commutator, Frattini, cosocle | C22×C8 | C25 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #60207 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
4737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13,17,37)(4,10)(5,23)(6,26)(7,8)(9,18,28,25)(11,15)(12,27)(14,40,42,38)(16,39,19,49)(20,29)(21,54)(22,55)(24,56)(30,44)(31,43,33,45)(32,48,34,51)(35,46)(36,53)(41,62,50,61)(47,57)(52,59)(58,63,60,64);(2,8)(3,14,17,42)(5,24)(9,31,28,33)(10,29)(12,26)(13,38,37,40)(15,44)(16,41,19,50)(18,45,25,43)(21,52)(32,58,34,60)(36,55)(39,61,49,62)(46,47)(48,64,51,63);(1,3,15,45,27,42,5,25,7,17,44,43,6,14,24,18)(2,9,23,40,26,31,11,37,8,28,56,38,12,33,30,13)(4,19,46,63,53,41,21,48,20,16,47,64,22,50,52,51)(10,34,54,61,55,60,35,39,29,32,59,62,36,58,57,49);(1,4)(2,10)(3,16)(5,21)(6,22)(7,20)(8,29)(9,32)(11,35)(12,36)(13,39)(14,41)(15,46)(17,19)(18,48)(23,54)(24,52)(25,51)(26,55)(27,53)(28,34)(30,57)(31,58)(33,60)(37,49)(38,61)(40,62)(42,50)(43,63)(44,47)(45,64)(56,59);(4,20)(10,29)(21,52)(22,53)(35,57)(36,55)(46,47)(54,59)Polycyclic presentation:
2^10|a^2=g,b^2=g,c^2=ghi,h^2=i,i^2=j,b^a=bj,c^a=ch,c^b=ci,d^c=dj,e^c=ef,e^d=egj,f^d=fj,g^c=gj,h^a=hi,h^b=hj,h^c=hj,i^a=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72307129, #72307980, #72308079, … |
| maximal normal subgroup of | #72307129, #72307980, #72308079, … |
| maximal quotient of | C2×#72307114 |
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.7);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.7*F.8*F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);