Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52570 |
| 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);(5,9)(8,10)(11,12)(13,16)(14,17)(15,20)(18,23)(19,21)(22,25)(24,26)(27,28,31,32)(29,33,35,38)(30,34,37,39)(36,40,41,42);(3,4)(5,8)(6,7)(9,10)(12,13)(14,21)(18,20)(22,26)(28,32)(29,35)(34,39)(36,41);(4,7)(8,10)(11,13,16,12)(14,17,21,19)(15,20,23,18)(22,24,26,25)(27,29)(28,33)(30,36)(31,35)(32,38)(34,40)(37,41)(39,42);(3,4)(5,8)(6,7)(9,10)(11,14,16,21)(12,17,13,19)(15,22,23,26)(18,24,20,25)(27,28)(29,33)(30,34)(31,32)(35,38)(36,40)(37,39)(41,42);(3,5)(4,8)(6,9)(7,10)(11,15,16,23)(12,18,13,20)(14,22,21,26)(17,24,19,25)(27,30,31,37)(28,34,32,39)(29,36,35,41)(33,40,38,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999641);