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×#52854 |
| 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 | |
4588 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)(14,22)(17,24)(20,26)(23,25)(27,28,33,36)(29,34,30,31)(32,40,41,35)(37,39,38,42);(3,4)(5,7)(6,8)(9,10)(11,12,15,18)(13,16,21,19)(14,17,22,24)(20,23,26,25)(27,29,33,30)(28,34,36,31)(32,37,41,38)(35,42,40,39);(4,5)(8,9)(11,12)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26)(27,30,33,29)(28,34,36,31)(32,38,41,37)(35,42,40,39);(3,5,7,4)(6,9,10,8)(11,13)(12,16)(14,20)(15,21)(17,23)(18,19)(22,26)(24,25)(27,31,33,34)(28,30,36,29)(32,39,41,42)(35,38,40,37);(3,6,7,10)(4,8,5,9)(11,14)(12,17)(13,20)(15,22)(16,23)(18,24)(19,25)(21,26)(27,32)(28,35)(29,37)(30,38)(31,39)(33,41)(34,42)(36,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999843);