Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#33784 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1676 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)(5,9)(6,8)(7,10)(11,12)(13,21)(14,17)(15,27)(16,30)(18,23)(19,29)(20,26)(22,37)(24,41)(25,38)(28,36)(31,35)(32,39)(33,40)(34,42);(3,5,6,10)(4,7,8,9)(12,17)(18,32)(19,33)(20,30)(21,37)(27,38)(34,36)(35,41);(5,10)(7,9)(11,13,14,22)(12,18,17,32)(15,28,25,42)(16,31,26,24)(19,35,33,41)(20,36,30,34)(21,38,37,27)(23,40,39,29);(4,8)(7,9)(12,17)(18,32)(19,33)(20,30)(21,37)(27,38)(34,36)(35,41)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530537);