Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11490 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | 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×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
982 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)(11,12,16,22,18,21,30,38,17,23,29,39,15,31,14,20)(13,24,27,34,26,19,33,35,28,37,41,42,32,36,25,40);(4,7)(5,9)(11,13)(12,19)(14,25)(15,32)(16,27)(17,28)(18,26)(20,34)(21,37)(22,35)(23,36)(24,31)(29,41)(30,33)(38,42)(39,40);(11,14,15,29,17,30,18,16)(12,20,31,39,23,38,21,22)(13,25,32,41,28,33,26,27)(19,34,24,40,36,42,37,35)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387347);