Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33652 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1636 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)(14,20)(15,28)(16,32)(17,22)(19,24)(21,30)(23,33)(25,36)(26,35)(27,39)(29,34)(31,42)(37,41)(38,40);(4,7)(8,10)(11,13)(12,18)(14,26)(15,24)(16,23)(17,25)(19,28)(20,35)(21,34)(22,36)(27,41)(29,30)(31,40)(32,33)(37,39)(38,42);(3,5,6,9)(4,8,7,10)(11,13)(12,19)(14,23)(15,29)(16,26)(17,25)(18,28)(20,34)(21,35)(22,38)(24,30)(27,40)(31,41)(32,37)(33,39)(36,42);(11,14,17,16)(12,20,22,32)(13,23,25,26)(15,27,31,30)(18,33,36,35)(19,34,38,37)(21,28,39,42)(24,29,40,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530239);