Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C25 |
| direct product | C2×#7167 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | D4 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2);(3,4)(5,6)(8,12)(9,10)(13,14)(15,16,18,23,20,17,22,26,21,25,19,24,29,28,30,27);(4,6)(7,8,10,13,11,14,9,12)(15,17,19,27,20,25,30,23,21,28,18,26,29,16,22,24)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33952);