Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62036 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(4,7)(5,9)(11,12,17,22)(13,18,25,33)(14,27,29,36)(15,21,16,20)(19,30,35,28)(23,37,39,42)(24,34,26,32)(31,40,41,38);(3,4)(5,8)(6,7)(9,10)(11,13)(12,18)(14,23)(15,24)(16,26)(17,25)(19,31)(20,32)(21,34)(22,33)(27,37)(28,38)(29,39)(30,40)(35,41)(36,42);(11,14,17,29)(12,19,22,35)(13,23,25,39)(15,28,16,30)(18,31,33,41)(20,27,21,36)(24,38,26,40)(32,37,34,42);(4,7)(8,10)(11,15,17,16)(12,20,22,21)(13,24,25,26)(14,28,29,30)(18,32,33,34)(19,27,35,36)(23,38,39,40)(31,37,41,42);(3,5,6,9)(4,8,7,10)(13,25)(18,33)(23,39)(24,26)(31,41)(32,34)(37,42)(38,40)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486024);