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×#33918 |
| 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)(13,17)(14,18)(15,28)(16,19)(20,25)(21,26)(22,30)(23,33)(24,40)(27,34)(29,36)(31,39)(32,37)(35,41)(38,42);(3,5)(4,8)(6,9)(7,10)(11,13)(12,17)(14,23)(15,24)(16,22)(18,33)(19,30)(20,31)(21,32)(25,39)(26,37)(27,36)(28,40)(29,34)(35,42)(38,41);(3,6,7,4)(5,9,10,8)(12,18)(13,22)(14,25)(15,26)(17,29)(19,34)(20,21)(23,38)(27,41)(28,35)(30,33)(31,40)(32,42)(36,39);(5,10)(8,9)(12,19)(17,30)(18,34)(20,35)(21,28)(29,33)(31,42)(32,40)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530999);