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×#26192 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2141 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)(5,8)(6,10)(7,9)(11,12)(13,23)(14,29)(15,32)(16,22)(17,21)(18,26)(19,31)(20,30)(24,37)(25,35)(27,39)(28,36)(33,38)(34,42)(40,41);(3,5,7,6)(4,8,9,10)(11,13,16,27)(12,18,22,36)(14,25,31,42)(15,28,17,26)(19,37,29,40)(20,35,38,34)(21,39,32,23)(24,33,41,30);(3,6,7,5)(4,8,9,10)(11,14)(12,19)(13,24)(15,30)(16,31)(17,33)(18,34)(20,21)(22,29)(23,40)(25,26)(27,41)(28,42)(32,38)(35,36)(37,39);(11,14)(12,20)(13,25)(15,33)(16,31)(17,30)(18,35)(19,21)(22,38)(23,40)(24,26)(27,42)(28,41)(29,32)(34,36)(37,39)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345792);