Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11492 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,5,8,6,7,9,10)(11,12,16,22,18,21,31,39,17,23,30,40,15,32,14,20)(13,24,27,38,26,19,25,35,28,37,41,34,33,36,29,42);(4,7)(5,9)(11,13)(12,19)(14,29)(15,33)(16,27)(17,28)(18,26)(20,38)(21,37)(22,35)(23,36)(24,32)(25,31)(30,41)(34,39)(40,42);(11,14,15,30,17,31,18,16)(12,20,32,40,23,39,21,22)(13,25,33,27,28,29,26,41)(19,34,24,35,36,38,37,42)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387349);