Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52661 |
| 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 | |
4566 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);(5,9)(8,10)(11,12)(13,16)(14,17)(15,20)(18,23)(19,21)(22,25)(24,26)(27,28,31,34)(29,35,37,32)(30,33,38,40)(36,41,42,39);(3,4)(5,8)(6,7)(9,10)(12,13)(14,21)(18,20)(22,26)(27,28,31,34)(29,32,37,35)(30,33,38,40)(36,39,42,41);(4,7)(8,10)(11,13,16,12)(14,17,21,19)(15,20,23,18)(22,24,26,25)(27,28)(29,32)(30,33)(31,34)(35,37)(36,39)(38,40)(41,42);(3,4)(5,8)(6,7)(9,10)(11,14,16,21)(12,17,13,19)(15,22,23,26)(18,24,20,25)(27,29)(28,32)(30,36)(31,37)(33,39)(34,35)(38,42)(40,41);(3,5)(4,8)(6,9)(7,10)(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, 7999433);