Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×#2183 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C23 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C23 | C23×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4);(5,6)(7,12)(8,10)(9,11)(13,14,16,20,17,15,22,23,18,21,25,28,26,24,19,27);(5,7,9,8)(6,10,11,12);(5,8,9,7)(6,10,11,12)(13,15,16,23,17,21,22,28,18,24,25,27,26,14,19,20)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400455);