Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#11709 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×#102 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1260 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)(11,12,14,19,17,22,16,21)(13,23,26,37,27,40,24,39)(15,29,28,20,32,36,31,38)(18,30,34,25,35,41,33,42);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,24)(15,30)(16,26)(17,27)(19,33)(20,37)(21,34)(22,35)(23,36)(25,31)(28,42)(29,40)(32,41)(38,39);(11,14,17,16)(12,19,22,21)(13,24,27,26)(15,28,32,31)(18,33,35,34)(20,36,38,29)(23,39,40,37)(25,30,42,41)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388666);