Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26502 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,6,7)(5,9,10,8)(11,12)(13,24)(14,30)(15,33)(16,22)(17,23)(18,27)(19,26)(20,35)(21,34)(25,40)(28,38)(29,41)(31,37)(32,42)(36,39);(4,7)(8,9)(11,13,16,27,17,29,34,28)(12,18,21,24,23,38,22,41)(14,15,26,35,32,36,31,25)(19,39,30,20,37,33,42,40);(11,14,16,26,17,32,34,31)(12,19,21,30,23,37,22,42)(13,25,27,15,29,35,28,36)(18,20,24,33,38,40,41,39);(3,5,6,10)(4,8,7,9)(11,15,34,25,17,36,16,35)(12,20,22,39,23,40,21,33)(13,26,28,14,29,31,27,32)(18,37,41,30,38,19,24,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351916);