Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62442 |
| 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 | |
6796 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);(4,7)(5,9)(11,12,16,20)(13,17,22,30)(14,24,25,39)(15,27,28,19)(18,32,33,26)(21,34,35,42)(23,38,37,31)(29,40,41,36);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,21)(15,23)(16,22)(18,29)(19,31)(20,30)(24,34)(25,35)(26,36)(27,38)(28,37)(32,40)(33,41)(39,42);(11,14,15,26,16,25,28,32)(12,18,19,24,20,33,27,39)(13,21,23,36,22,35,37,40)(17,29,31,34,30,41,38,42);(4,7)(8,10)(14,25)(18,33)(21,35)(24,39)(26,32)(29,41)(34,42)(36,40);(3,5,6,9)(4,8,7,10)(13,22)(17,30)(21,35)(23,37)(29,41)(31,38)(34,42)(36,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486405);