Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,8)(9,10,13,14)(11,16,15,12)(17,18)(19,23)(20,21)(22,24);(1,2,4,3)(10,14)(11,15)(17,18,21,20)(19,23,24,22);(2,3)(10,14)(12,16)(18,20)(22,23);(5,6)(7,8)(9,11,13,15)(10,12,14,16)(17,19)(18,22)(20,23)(21,24);(1,3,4,2)(9,12)(10,15)(11,14)(13,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^9|a^2=fh,b^2=h,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280871, #72280903, #72281037, … |
| maximal normal subgroup of | #72280871, #72280903, #72281037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 912282);