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×#11978 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#466 | C4 |
| 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 | |
800 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)(11,12,14,20,28,34,41,36,15,18,29,19,16,21,30,38)(13,22,24,39,31,37,27,32,25,40,23,33,26,35,42,17);(3,4)(5,8)(6,10)(7,9)(11,13)(12,17)(14,27)(15,25)(16,31)(18,32)(19,35)(20,37)(21,39)(22,36)(23,30)(24,41)(26,28)(29,42)(33,34)(38,40);(3,5,7,6)(4,8,9,10)(12,18)(17,32)(19,20)(21,34)(22,40)(33,39)(35,37)(36,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390351);