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×#33922 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1352 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)(11,12,14,19)(13,22,23,17)(15,18,26,34)(16,29,27,21)(20,36,35,28)(24,30,39,33)(25,32,40,38)(31,42,37,41);(3,5)(4,8)(6,9)(7,10)(11,13)(12,17)(14,23)(15,24)(16,25)(18,33)(19,22)(20,31)(21,32)(26,39)(27,40)(28,42)(29,38)(30,34)(35,37)(36,41);(3,6,7,4)(5,9,10,8)(12,18)(13,23)(15,28)(16,27)(17,30)(19,34)(20,21)(22,33)(24,41)(26,36)(29,35)(31,38)(32,37)(39,42);(5,10)(8,9)(12,19)(13,23)(18,34)(20,35)(21,29)(24,39)(25,40)(41,42)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531003);