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×#54917 |
| 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 | |
4582 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);(11,12,15,16)(13,19,20,17)(14,22,23,18)(21,24,26,25)(27,28,31,34)(29,35,37,32)(30,33,38,40)(36,41,42,39);(3,4,7,8)(5,6,9,10)(12,16)(13,20)(18,22)(21,26)(30,38)(33,40)(36,42)(39,41);(4,8)(5,9)(11,12)(13,17)(14,18)(15,16)(19,20)(21,24)(22,23)(25,26)(27,29,31,37)(28,32,34,35)(30,36,38,42)(33,39,40,41);(3,5)(4,6)(7,9)(8,10)(11,13,15,20)(12,17,16,19)(14,21,23,26)(18,24,22,25)(27,28)(29,35)(30,33)(31,34)(32,37)(36,41)(38,40)(39,42);(3,6)(4,5)(7,10)(8,9)(11,14,15,23)(12,18,16,22)(13,21,20,26)(17,24,19,25)(27,30)(28,33)(29,36)(31,38)(32,39)(34,40)(35,41)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008820);