Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#357 |
| commutator | C16 | C23×C8 |
| Frattini, cosocle | C4×C16 | C24 |
| prime radical | C2×#3415 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ C4×#504 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,4,7)(3,5,8,6)(11,25)(12,31)(13,38)(14,33)(15,16)(19,42)(20,47)(21,53)(22,49)(23,24)(26,36)(27,59)(28,58)(29,57)(30,32)(34,37)(35,60)(39,63)(40,48)(41,50)(43,52)(44,69)(45,68)(46,67)(51,64)(54,72)(55,56)(61,62)(65,66)(70,71);(9,10)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,18)(25,42)(26,43)(27,44)(28,45)(29,46)(30,40)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,41)(38,53)(39,54)(55,65)(56,66)(57,67)(58,68)(59,69)(60,64)(61,70)(62,71)(63,72);(10,18)(19,40)(20,41)(21,22)(23,24)(42,48)(43,64)(44,54)(45,46)(47,50)(49,53)(51,52)(65,70)(66,71)(67,68)(69,72);(2,5,7,6)(9,11,26,59,13,39,12,32,15,28,34,56,33,55,35,58,17,30,60,63,14,27,37,25,16,29,31,62,38,61,36,57)(10,19,43,69,21,54,20,48,23,45,50,66,49,65,51,68,18,40,64,72,22,44,41,42,24,46,47,71,53,70,52,67)Polycyclic presentation:
2^10|a^2=hi,d^2=efhj,e^2=gi,f^2=h,g^2=ij,h^2=j,i^2=j,c^b=cj,d^a=de,e^a=eg,e^d=ej,f^a=fi,f^d=fj,g^a=gi,h^a=hj,i^a=ij184 conjugacy classes — view the full character table.
| maximal subgroup of | #72232013, #72322316, #72322477 |
| maximal normal subgroup of | #72232013, #72322316, #72322477 |
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.8*F.9);;
gap> SetPower(coll, 4, F.5*F.6*F.8*F.10);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);