Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11935 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
774 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);(4,7)(5,9)(8,10)(11,12,15,21,16,22,30,36,17,19,31,37,29,38,14,20)(13,23,25,33,26,39,24,40,27,41,28,18,32,34,42,35);(3,4,6,8)(5,10,9,7)(11,13)(12,18)(14,28)(15,24)(16,32)(17,27)(19,33)(20,39)(21,23)(22,40)(25,30)(26,29)(31,42)(34,36)(35,38)(37,41);(12,19)(18,33)(20,36)(21,37)(22,38)(23,41)(34,39)(35,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390250);