Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13212 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,6)(7,8,11,16,13,17,12,28)(9,18,20,35,21,34,27,36)(10,22,24,30,26,38,25,15)(14,29,31,19,33,37,32,23);(7,9,10,23,12,20,25,29,13,21,26,19,11,27,24,37)(8,14,30,18,16,32,38,36,17,33,15,34,28,31,22,35);(3,4,5,6)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402332);