Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33735 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C22×D8 |
| prime radical | C2×#2215 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1454 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)(11,12)(13,18)(14,24)(15,29)(16,31)(17,21)(19,26)(20,25)(22,35)(23,34)(27,41)(28,38)(30,42)(32,36)(33,40)(37,39);(4,7)(8,10)(11,13,17,16)(12,18,21,31)(14,22,28,27)(15,23,30,25)(19,32,33,37)(20,29,34,42)(24,35,38,41)(26,36,40,39);(11,14,25,39,16,27,30,40,17,28,23,36,13,22,15,26)(12,19,29,35,18,32,34,38,21,33,42,41,31,37,20,24);(3,5,6,9)(4,8,7,10)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530422);