Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#57630 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4636 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);(3,4)(5,8)(6,7)(9,10)(11,12,16,13)(14,17,21,19)(15,20,23,18)(22,25,26,24)(27,28)(29,35)(30,38)(31,37)(32,33)(34,39)(36,41)(40,42);(3,5,7,10)(4,8,6,9)(12,13)(14,21)(15,23)(24,25)(28,33)(30,39)(31,35)(36,40);(4,6)(8,9)(11,13,16,12)(14,17,21,19)(15,20,23,18)(22,24,26,25)(27,29)(28,31)(30,40)(32,37)(33,35)(34,42)(36,39)(38,41);(3,6,7,4)(5,9,10,8)(11,14,16,21)(12,17,13,19)(15,22,23,26)(18,24,20,25)(27,30)(28,34)(29,36)(31,41)(32,39)(33,38)(35,42)(37,40);(3,5)(4,8)(6,9)(7,10)(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,31)(28,29)(30,41)(32,35)(33,37)(34,36)(38,40)(39,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008611);