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×#56162 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2828 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4810 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);(4,7)(5,9)(11,12,15,18)(13,19,21,16)(14,17,22,24)(20,25,26,23)(27,28,31,32)(29,35,36,33)(30,38,39,34)(37,40,42,41);(3,4,6,7)(5,10,9,8)(14,22)(17,24)(20,26)(23,25)(28,32)(29,36)(34,38)(37,42);(3,4,6,7)(5,8,9,10)(11,12)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26)(27,29,31,36)(28,33,32,35)(30,37,39,42)(34,40,38,41);(3,4)(5,8)(6,7)(9,10)(11,13,15,21)(12,16,18,19)(14,20,22,26)(17,23,24,25)(27,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,39)(41,42);(3,5)(4,8)(6,9)(7,10)(11,14,15,22)(12,17,18,24)(13,20,21,26)(16,23,19,25)(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, 8008824);