Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,24)(19,22,27,31)(20,23,28,32)(25,29,26,30);(9,11,12,16)(10,13,14,15)(17,19,20,26,21,27,28,25)(18,22,23,30,24,31,32,29);(1,2,3,4)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31);(5,7,8,6)(10,14)(13,15)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=f,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286110);