Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#45853 |
| 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 | |
4502 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);(3,4,6,8)(5,7,9,10)(13,19)(17,22)(20,25)(24,26)(27,28,32,33)(29,30,35,38)(31,39,40,34)(36,42,41,37);(12,16)(13,19)(14,21)(24,26)(28,33)(29,35)(31,40)(37,42);(3,4)(5,7)(6,8)(9,10)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,29)(28,30)(31,36)(32,35)(33,38)(34,37)(39,42)(40,41);(3,5,6,9)(4,7,8,10)(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,30)(28,29)(31,37)(32,38)(33,35)(34,36)(39,41)(40,42);(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,31,32,40)(28,34,33,39)(29,36,35,41)(30,37,38,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609575);