Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53220 |
| 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 | |
4620 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);(5,9)(8,10)(11,12)(13,17)(14,16)(15,22)(18,23)(19,20)(21,26)(24,25)(27,28,32,35)(29,30,37,33)(31,39,40,34)(36,41,42,38);(4,7)(8,10)(12,14)(13,19)(15,23)(24,26)(27,29,32,37)(28,33,35,30)(31,36,40,42)(34,41,39,38);(3,4)(5,8)(6,7)(9,10)(11,13,16,19)(12,17,14,20)(15,21,23,25)(18,24,22,26)(27,29)(28,33)(30,35)(31,36)(32,37)(34,41)(38,39)(40,42);(3,5)(4,8)(6,9)(7,10)(11,14,16,12)(13,20,19,17)(15,22,23,18)(21,26,25,24)(27,30,32,33)(28,29,35,37)(31,38,40,41)(34,36,39,42);(3,4)(5,8)(6,7)(9,10)(11,15)(12,18)(13,21)(14,22)(16,23)(17,24)(19,25)(20,26)(27,31)(28,34)(29,36)(30,38)(32,40)(33,41)(35,39)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012971);