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×#57777 |
| 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 | |
4566 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)(6,9)(12,16)(13,19)(18,23)(20,25)(27,28,33,35)(29,36,30,31)(32,40,41,34)(37,39,38,42);(3,4)(5,9)(6,10)(7,8)(12,16)(17,22)(18,23)(24,26)(27,29,33,30)(28,31,35,36)(32,37,41,38)(34,39,40,42);(4,8)(5,10)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,30,33,29)(28,31,35,36)(32,38,41,37)(34,39,40,42);(3,5)(4,6)(7,10)(8,9)(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,31)(28,29)(30,35)(32,39)(33,36)(34,37)(38,40)(41,42);(3,6)(4,5)(7,9)(8,10)(11,14,15,21)(12,18,16,23)(13,20,19,25)(17,24,22,26)(27,32)(28,34)(29,37)(30,38)(31,39)(33,41)(35,40)(36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009079);