Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33917 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3339 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1320 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);(3,4,7,6)(5,9,10,8)(11,12)(13,22)(14,19)(15,27)(16,31)(17,21)(18,25)(20,29)(23,38)(24,40)(26,37)(28,34)(30,39)(32,36)(33,41)(35,42);(3,5)(4,8)(6,9)(7,10)(13,23)(18,33)(22,38)(24,32)(25,41)(28,42)(34,35)(36,40);(3,6,7,4)(5,9,10,8)(11,13,17,23)(12,18,21,33)(14,24,16,32)(15,28,30,42)(19,34,31,35)(20,36,37,40)(22,39,38,27)(25,26,41,29);(5,10)(8,9)(11,14,17,16)(12,19,21,31)(13,24,23,32)(15,26,30,29)(18,34,33,35)(20,27,37,39)(22,40,38,36)(25,28,41,42)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530997);