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×#58832 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2825 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4842 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)(5,7)(6,9)(8,10)(11,12,15,18)(13,19,21,16)(14,17,22,24)(20,25,26,23)(27,28)(29,32)(30,38)(31,34)(33,39)(35,37)(36,41)(40,42);(4,5)(6,10)(14,22)(17,24)(20,26)(23,25)(27,28,31,34)(29,35,37,32)(30,38,39,33)(36,40,42,41);(4,5)(8,9)(11,13,15,21)(12,16,18,19)(14,20,22,26)(17,23,24,25)(27,28)(29,35)(30,33)(31,34)(32,37)(36,41)(38,39)(40,42);(3,5,7,4)(6,9,10,8)(11,12)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26)(27,29)(28,32)(30,36)(31,37)(33,40)(34,35)(38,41)(39,42);(3,6,7,10)(4,8,5,9)(11,14,15,22)(12,17,18,24)(13,20,21,26)(16,23,19,25)(27,30)(28,33)(29,36)(31,39)(32,40)(34,38)(35,41)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012298);