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×#56997 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2635 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4794 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);(3,4,7,5)(6,9,10,8)(11,12)(13,17)(14,18)(15,16)(19,21)(20,25)(22,24)(23,26)(28,29)(30,37)(31,39)(40,41);(4,5)(8,9)(12,15)(13,19)(14,22)(25,26)(27,28,32,29)(30,35,37,33)(31,36,39,34)(38,40,42,41);(11,13,16,19)(12,17,15,21)(14,23,22,20)(18,26,24,25)(27,29,32,28)(30,33,37,35)(31,36,39,34)(38,40,42,41);(3,5,7,4)(6,9,10,8)(11,14,16,22)(12,18,15,24)(13,20,19,23)(17,25,21,26)(27,30,32,37)(28,33,29,35)(31,38,39,42)(34,40,36,41);(3,6,7,10)(4,8,5,9)(11,15,16,12)(13,21,19,17)(14,24,22,18)(20,26,23,25)(27,31)(28,34)(29,36)(30,38)(32,39)(33,40)(35,41)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006952);