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×#54244 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4674 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)(6,10)(12,16)(13,19)(18,23)(21,25)(27,28,32,36)(29,33,37,42)(30,34,40,39)(31,35,41,38);(3,4,7,8)(5,10,9,6)(14,22)(18,23)(21,25)(24,26)(29,37)(31,41)(33,42)(35,38);(4,8)(5,9)(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24)(27,29)(28,33)(30,35)(31,39)(32,37)(34,41)(36,42)(38,40);(3,5,7,9)(4,6,8,10)(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,30)(28,34)(29,38)(31,33)(32,40)(35,37)(36,39)(41,42);(3,6)(4,9)(5,8)(7,10)(11,14)(12,18)(13,21)(15,22)(16,23)(17,24)(19,25)(20,26)(27,31)(28,35)(29,39)(30,33)(32,41)(34,37)(36,38)(40,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008765);