Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61646 |
| 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 | |
7562 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)(5,9)(6,8)(7,10)(11,12)(13,17)(14,24)(15,27)(16,19)(18,25)(20,29)(21,35)(22,31)(23,39)(26,34)(28,33)(30,37)(32,38)(36,42)(40,41);(3,5)(4,7)(6,10)(8,9)(11,13,16,22)(12,17,19,31)(14,21,26,36)(15,23,29,38)(18,30,33,41)(20,32,27,39)(24,35,34,42)(25,37,28,40);(11,14,16,26)(12,18,19,33)(13,21,22,36)(15,28,29,25)(17,30,31,41)(20,34,27,24)(23,40,38,37)(32,42,39,35);(5,10)(7,9)(12,19)(13,22)(18,33)(20,27)(21,36)(23,38)(24,34)(37,40);(4,8)(7,9)(13,22)(17,31)(21,36)(23,38)(30,41)(32,39)(35,42)(37,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484445);