Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33686 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3176 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(12,17)(13,21)(15,26)(16,25)(18,32)(19,35)(22,40)(23,41)(24,39)(27,28)(29,33)(30,37)(34,36)(38,42);(4,7)(8,10)(11,12)(13,18)(14,17)(15,19)(16,20)(21,33)(22,29)(23,34)(24,30)(25,31)(26,37)(27,36)(28,38)(32,40)(35,39)(41,42);(11,13,14,22)(12,18,17,29)(15,27,24,41)(16,28,25,23)(19,36,30,42)(20,38,31,34)(21,39,40,26)(32,37,33,35);(3,5,6,9)(4,8,7,10)(12,17)(18,29)(19,30)(20,31)(32,33)(34,38)(35,37)(36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530289);