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×#54311 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4452 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)(5,9)(6,7)(8,10)(11,12,16,17)(13,18,21,24)(14,19,23,26)(15,22,25,20)(28,29)(30,37)(31,39)(40,41);(4,7)(8,9)(12,17)(13,21)(14,23)(15,25)(27,28,32,29)(30,35,37,33)(31,36,39,34)(38,40,42,41);(11,13)(12,18)(14,20)(15,19)(16,21)(17,24)(22,23)(25,26)(27,29,32,28)(30,33,37,35)(31,36,39,34)(38,40,42,41);(3,4)(5,8)(6,7)(9,10)(11,14)(12,19)(13,22)(15,24)(16,23)(17,26)(18,25)(20,21)(27,30,32,37)(28,33,29,35)(31,38,39,42)(34,40,36,41);(3,5,6,10)(4,8,7,9)(11,15,16,25)(12,20,17,22)(13,19,21,26)(14,24,23,18)(27,31,32,39)(28,34,29,36)(30,38,37,42)(33,40,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010968);