Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C24×D4 |
| commutator | C2×C4 | C25×C4 |
| Frattini, cosocle | C22×C4 | C26 |
| socle | C22 | C24×C22⋊C4 |
| prime radical | C22×D4×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
41389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(6,8)(9,10,14,13)(11,17,19,15)(12,16,21,22)(18,24,23,20);(9,11)(10,15)(12,20)(13,17)(14,19)(16,23)(18,22)(21,24);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18,19,23)(13,22)(14,21)(15,20,17,24);(2,5)(6,8)(12,21)(16,22)(18,23)(20,24);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^10|a^2=j,c^2=h,f^2=ij,g^2=j,b^a=bj,c^b=cg,d^c=di,e^c=eij,f^a=fij,g^b=gj,h^b=hj196 conjugacy classes — view the full character table.
| maximal subgroup of | #72239784, #72249657, #72249704, … |
| maximal normal subgroup of | #72239784, #72249657, #72249704, … |
| maximal quotient of | #72264662 |
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.10);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);