Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26925 |
| 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 | |
2081 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)(11,12,16,21,18,14,17,22)(13,23,26,29,28,24,27,19)(15,31,32,30,34,36,33,20)(25,35,39,38,41,42,40,37);(4,7)(8,9)(11,13,18,28)(12,19,14,29)(15,25,34,41)(16,27,17,26)(20,35,30,42)(21,24,22,23)(31,37,36,38)(32,40,33,39);(11,14,18,12)(13,24,28,23)(15,30,34,20)(16,21,17,22)(19,26,29,27)(25,38,41,37)(31,33,36,32)(35,40,42,39);(3,5,6,10)(4,8,7,9)(11,15)(12,20)(13,25)(14,30)(16,33)(17,32)(18,34)(19,35)(21,36)(22,31)(23,37)(24,38)(26,40)(27,39)(28,41)(29,42)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346926);