Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61207 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12)(13,18)(14,20)(15,21)(16,30)(17,22)(19,26)(23,32)(24,33)(25,35)(27,36)(28,34)(29,37)(31,39)(38,41)(40,42);(3,5)(4,7)(6,10)(8,9)(11,13,17,25)(12,18,22,35)(14,23,27,40)(15,28,16,24)(19,31,37,41)(20,32,36,42)(21,34,30,33)(26,39,29,38);(11,14)(12,19)(13,23)(15,26)(16,29)(17,27)(18,31)(20,30)(21,36)(22,37)(24,38)(25,40)(28,39)(32,33)(34,42)(35,41);(5,10)(7,9)(11,14)(12,20)(13,23)(15,29)(16,26)(17,27)(18,32)(19,30)(21,37)(22,36)(24,39)(25,40)(28,38)(31,33)(34,41)(35,42);(11,15,17,16)(12,21,22,30)(13,24,25,28)(14,26,27,29)(18,33,35,34)(19,36,37,20)(23,38,40,39)(31,32,41,42)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483565);