Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11984 |
| 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 | |
784 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);(4,8)(5,6)(9,10)(11,12,15,21,17,22,30,20,16,19,31,36,29,37,14,28)(13,23,25,33,27,38,41,40,26,39,42,18,32,34,24,35);(3,4)(5,8)(6,10)(7,9)(11,13,16,26)(12,18,19,33)(14,25,30,42)(15,24,31,41)(17,32,29,27)(20,38,28,34)(21,39,36,23)(22,40,37,35);(3,5,7,6)(4,8,9,10)(12,19)(18,33)(20,28)(21,36)(22,37)(23,39)(34,38)(35,40)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390365);