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×#62090 |
| 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 | |
6488 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)(11,12,16,21)(13,23,26,18)(14,28,30,36)(15,22,17,20)(19,31,35,29)(24,38,40,42)(25,33,27,34)(32,39,37,41);(3,4)(5,8)(6,7)(9,10)(11,13,16,26)(12,18,21,23)(14,24,30,40)(15,25,17,27)(19,32,35,37)(20,33,22,34)(28,42,36,38)(29,39,31,41);(11,14)(12,19)(13,24)(15,29)(16,30)(17,31)(18,32)(20,28)(21,35)(22,36)(23,37)(25,39)(26,40)(27,41)(33,42)(34,38);(4,7)(8,10)(11,15,16,17)(12,20,21,22)(13,25,26,27)(14,29,30,31)(18,33,23,34)(19,28,35,36)(24,39,40,41)(32,42,37,38);(3,5,6,9)(4,8,7,10)(12,21)(13,26)(19,35)(20,22)(24,40)(25,27)(28,36)(39,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486084);