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×#61198 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ 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)(13,23)(14,27)(15,20)(16,22)(17,21)(18,26)(19,28)(24,38)(25,34)(29,35)(30,33)(31,36)(32,39)(37,41)(40,42);(3,5)(4,7)(6,10)(8,9)(11,13,16,26)(12,18,22,23)(14,24,29,40)(15,30,17,25)(19,32,36,37)(20,34,21,33)(27,42,35,38)(28,41,31,39);(11,14)(12,19)(13,24)(15,28)(16,29)(17,31)(18,32)(20,35)(21,27)(22,36)(23,37)(25,39)(26,40)(30,41)(33,42)(34,38);(5,10)(7,9)(11,15,16,17)(12,20,22,21)(13,25,26,30)(14,28,29,31)(18,33,23,34)(19,35,36,27)(24,39,40,41)(32,42,37,38);(11,15,16,17)(12,21,22,20)(13,25,26,30)(14,28,29,31)(18,34,23,33)(19,27,36,35)(24,39,40,41)(32,38,37,42)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483549);