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×#58748 |
| 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);(3,4,6,8)(5,9,10,7)(11,12)(13,16)(14,19)(15,20)(17,21)(18,23)(22,24)(25,26)(27,28,33,36)(29,34,30,31)(32,35,40,42)(37,41,38,39);(5,10)(7,9)(12,13)(14,21)(18,20)(22,26)(27,29,33,30)(28,34,36,31)(32,38,40,37)(35,39,42,41);(3,5)(4,7)(6,10)(8,9)(11,13,16,12)(14,19,21,17)(15,20,23,18)(22,25,26,24)(27,30,33,29)(28,34,36,31)(32,38,40,37)(35,41,42,39);(3,4)(5,7)(6,8)(9,10)(11,14)(12,17)(13,19)(15,22)(16,21)(18,24)(20,25)(23,26)(27,31,33,34)(28,30,36,29)(32,39,40,41)(35,38,42,37);(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,32)(28,35)(29,37)(30,38)(31,39)(33,40)(34,41)(36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012135);