Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#47322 |
| 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 | |
4648 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);(4,7)(8,10)(11,12)(13,17)(14,22)(15,16)(18,23)(19,20)(21,26)(24,25)(27,28,31,34)(29,32,37,35)(30,33,39,38)(36,40,42,41);(12,16)(13,19)(14,23)(24,26)(27,28,31,34)(29,32,37,35)(30,38,39,33)(36,41,42,40);(3,4)(5,8)(6,7)(9,10)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,28)(29,35)(30,33)(31,34)(32,37)(36,41)(38,39)(40,42);(11,13,15,19)(12,17,16,20)(14,21,23,25)(18,24,22,26)(27,29)(28,32)(30,36)(31,37)(33,40)(34,35)(38,41)(39,42);(3,5,6,9)(4,8,7,10)(11,14)(12,18)(13,21)(15,23)(16,22)(17,24)(19,25)(20,26)(27,30,31,39)(28,33,34,38)(29,36,37,42)(32,40,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608504);