Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#11706 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C25 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12)(13,23,26,39)(14,19)(15,29)(16,21)(17,22)(18,25,34,42)(20,28)(24,37,27,40)(30,33,41,35)(31,38)(32,36);(3,5)(4,7)(6,10)(8,9)(11,13,17,26)(12,18,22,34)(14,27,16,24)(15,30,32,41)(19,35,21,33)(20,37,38,40)(23,29,39,36)(25,28,42,31);(11,14,17,16)(12,19,22,21)(13,24,26,27)(15,28,32,31)(18,33,34,35)(20,36,38,29)(23,37,39,40)(25,41,42,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388659);