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×#62337 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3551 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6494 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)(11,12,17,21)(13,18,24,35)(14,25,29,41)(15,20,16,31)(19,27,37,28)(22,38,26,40)(23,33,30,34)(32,42,36,39);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,26)(15,30)(16,23)(17,24)(19,36)(20,34)(21,35)(22,29)(25,40)(27,39)(28,42)(31,33)(32,37)(38,41);(11,14,16,28,17,29,15,27)(12,19,20,25,21,37,31,41)(13,22,23,39,24,26,30,42)(18,32,34,38,35,36,33,40);(5,10)(7,9)(11,13)(12,18)(14,22)(15,30)(16,23)(17,24)(19,32)(20,34)(21,35)(25,38)(26,29)(27,42)(28,39)(31,33)(36,37)(40,41);(4,8)(7,9)(11,15,17,16)(12,20,21,31)(13,23,24,30)(14,27,29,28)(18,33,35,34)(19,25,37,41)(22,39,26,42)(32,40,36,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484825);