Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #24196 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#33595 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20436 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,12)(7,17,21,22)(8,23)(13,16)(14,18,27,32)(19,34,25,35)(20,26)(29,31,36,33);(1,2,3,4)(5,6,10,13,9,12,11,16)(7,18,22,14,21,32,17,27)(8,24,26,30,23,15,20,28)(19,36,34,33,25,29,35,31);(6,13)(7,19,21,25)(8,20)(10,11)(12,16)(14,29,27,36)(17,34,22,35)(18,31,32,33)(23,26)(28,30);(5,7,20,35,10,17,8,25,9,21,26,34,11,22,23,19)(6,14,30,29,13,18,15,33,12,27,28,36,16,32,24,31)Polycyclic presentation:
2^10|a^2=g,b^2=fi,c^2=g,d^2=egi,e^2=i,h^2=j,i^2=j,b^a=bf,c^b=ci,d^a=dh,d^b=dij,d^c=de,e^a=ej,e^b=ej,e^c=ei,e^d=ej,f^d=fj,g^d=gj,h^b=hj,h^c=hj,i^c=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72250694, #72251004, #72252721, … |
| maximal normal subgroup of | #72250694, #72251004, #72252721, … |
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.6*F.9);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.5*F.7*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);