Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #10995 |
| commutator | C22×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4×D4 | C23 |
| prime radical | #876092 | C2 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
2684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(15,20)(17,23);(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,18,22,24)(19,23);(3,8)(5,7)(9,11,13,19)(10,14,16,22)(12,18,21,24)(15,17,20,23)Polycyclic presentation:
2^10|a^2=g,b^2=e,c^2=j,f^2=i,b^a=bf,c^a=cg,c^b=cd,d^a=dh,d^b=dh,e^c=eh,e^d=ej,f^b=fi,f^c=fhij,f^d=fj,g^b=gi,h^b=hj76 conjugacy classes — view the full character table.
| maximal subgroup of | #72234059, #72234085, #72234090, … |
| maximal normal subgroup of | #72234059, #72234085, #72234090, … |
| maximal quotient of | C2×#72234058, #72261334 |
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.5);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);