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×#57420 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2688 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4826 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);(11,12)(13,17)(14,22)(15,16)(18,23)(19,20)(21,26)(24,25)(27,28,31,32)(29,35,36,33)(30,38,39,34)(37,40,42,41);(3,4,7,9)(5,6,10,8)(12,16)(13,19)(14,23)(24,26)(28,32)(29,36)(34,38)(37,42);(3,5,7,10)(4,8,9,6)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,29,31,36)(28,33,32,35)(30,37,39,42)(34,40,38,41);(3,5)(4,6)(7,10)(8,9)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,26)(23,25)(27,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,39)(41,42);(3,6)(4,5)(7,8)(9,10)(11,14,15,23)(12,18,16,22)(13,21,19,25)(17,24,20,26)(27,30)(28,34)(29,37)(31,39)(32,38)(33,40)(35,41)(36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007722);