Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#28823 |
| 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 | |
1734 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);(3,4,6,8)(5,9,10,7)(11,12)(13,19)(14,26)(15,18)(16,21)(17,22)(20,27)(23,31)(24,38)(25,33)(28,36)(29,32)(30,35)(34,37)(39,41)(40,42);(12,18,22,21)(13,23,25,30)(19,33)(20,32,36,26)(24,37,39,40)(31,35)(34,42)(38,41);(3,5,6,10)(4,7,8,9)(11,13)(12,19)(14,24)(15,23)(16,30)(17,25)(18,31)(20,34)(21,35)(22,33)(26,38)(27,37)(28,40)(29,39)(32,41)(36,42);(11,14,15,27,17,29,16,28)(12,20,21,26,22,36,18,32)(13,24,23,37,25,39,30,40)(19,34,35,38,33,42,31,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372979);