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×#11924 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×#102 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1140 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)(13,23,26,39)(14,19)(15,29)(16,21)(17,22)(18,25,34,42)(20,31)(24,36,27,40)(28,38)(30,33,41,35)(32,37);(4,7)(5,9)(11,13)(12,18)(14,24)(15,30)(16,27)(17,26)(19,33)(20,36)(21,35)(22,34)(23,37)(25,28)(29,39)(31,42)(32,41)(38,40);(11,14,17,16)(12,19,22,21)(13,24,26,27)(15,28,32,31)(18,33,34,35)(20,29,38,37)(23,36,39,40)(25,41,42,30)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390234);