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×#46217 |
| 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 | |
4518 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);(4,8)(9,10)(11,12,16,14)(13,19,21,17)(15,22,23,18)(20,24,26,25)(27,28)(29,35)(30,36)(31,39)(32,33)(34,40)(37,38)(41,42);(12,14)(17,19)(18,22)(24,25)(28,33)(29,36)(31,40)(38,41);(3,4,7,8)(5,9,6,10)(11,13)(12,17)(14,19)(15,20)(16,21)(18,24)(22,25)(23,26)(27,29)(28,30)(31,37)(32,36)(33,35)(34,38)(39,41)(40,42);(3,5,7,6)(4,9,8,10)(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,30)(28,29)(31,38)(32,35)(33,36)(34,37)(39,42)(40,41);(3,6,7,5)(4,10,8,9)(11,15,16,23)(12,18,14,22)(13,20,21,26)(17,24,19,25)(27,31,32,40)(28,34,33,39)(29,37,36,42)(30,38,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609944);