Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11987 |
| 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 | |
768 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)(11,12,15,21,29,37,30,39,17,19,14,20,16,22,32,41)(13,23,25,36,33,40,28,18,27,35,24,42,26,38,31,34);(3,4)(5,8)(6,10)(7,9)(11,13,17,27)(12,18,19,34)(14,28,15,31)(16,33,29,26)(20,38,21,40)(22,42,37,36)(23,41,35,39)(24,32,25,30);(3,5,7,6)(4,8,9,10)(12,19)(18,34)(20,21)(22,37)(23,35)(36,42)(38,40)(39,41)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390368);