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×#56038 |
| 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 | |
4810 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)(6,8)(7,10)(11,12)(13,16)(14,19)(15,20)(17,21)(18,23)(22,24)(25,26)(27,28,31,34)(29,35,37,32)(30,33,38,40)(36,41,42,39);(3,4,6,8)(5,7,10,9)(12,13)(14,21)(18,20)(22,26)(30,38)(33,40)(36,42)(39,41);(11,13,16,12)(14,19,21,17)(15,20,23,18)(22,25,26,24)(27,29,31,37)(28,32,34,35)(30,36,38,42)(33,39,40,41);(3,4)(5,7)(6,8)(9,10)(11,14)(12,17)(13,19)(15,22)(16,21)(18,24)(20,25)(23,26)(27,28)(29,35)(30,33)(31,34)(32,37)(36,41)(38,40)(39,42);(3,5)(4,7)(6,10)(8,9)(11,15,16,23)(12,18,13,20)(14,22,21,26)(17,24,19,25)(27,30,31,38)(28,33,34,40)(29,36,37,42)(32,39,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007068);