Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #21408 |
| commutator | C2×C8 | C22×C42 |
| Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#3060 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#2875 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(7,16,19,20)(8,21)(9,24)(10,23)(12,27,29,17)(13,25)(14,26)(18,31,32,33)(22,30)(28,36,35,34);(2,4)(5,6)(7,17)(8,22)(9,25)(10,26)(11,15)(12,20)(13,24)(14,23)(16,29)(18,34)(19,27)(21,30)(28,33)(31,35)(32,36);(7,18,19,32)(8,9)(12,28,29,35)(13,30)(16,31,20,33)(17,34,27,36)(21,24)(22,25);(5,7,9,20,10,18,24,31,11,19,8,16,23,32,21,33)(6,12,13,27,14,28,25,34,15,29,30,17,26,35,22,36)Polycyclic presentation:
2^10|a^2=f,c^2=fh,d^2=g,e^2=i,g^2=i,i^2=j,b^a=bf,d^a=de,d^b=dej,d^c=di,e^a=ei,e^b=eij,e^c=ej,f^d=fj,g^a=gi,g^b=gi,g^c=gj,i^a=ij,i^b=ij130 conjugacy classes — view the full character table.
| maximal subgroup of | #72250103, #72256914, #72257024, … |
| maximal normal subgroup of | #72250103, #72256914, #72257024, … |
| maximal quotient of | #72265505 |
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.6);;
gap> SetPower(coll, 3, F.6*F.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);