Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41396 |
| 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 | |
4664 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)(7,10)(11,12,16,17)(13,14,19,22)(15,18,23,24)(20,21,25,26)(27,28)(29,34)(30,31)(32,33)(35,40)(36,42)(37,39)(38,41);(3,4,6,8)(5,7,9,10)(12,17)(13,19)(18,24)(20,25)(28,33)(31,35)(34,41)(37,42);(3,5)(4,7)(6,9)(8,10)(11,13)(12,14)(15,20)(16,19)(17,22)(18,21)(23,25)(24,26)(27,29)(28,34)(30,39)(31,37)(32,38)(33,41)(35,42)(36,40);(11,14)(12,13)(15,21)(16,22)(17,19)(18,20)(23,26)(24,25)(27,30)(28,35)(29,36)(31,33)(32,40)(34,37)(38,39)(41,42);(11,15,16,23)(12,18,17,24)(13,20,19,25)(14,21,22,26)(27,31,32,35)(28,30,33,40)(29,37,38,42)(34,39,41,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547085);