Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#45778 |
| 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)(13,19)(17,22)(20,25)(24,26)(27,28)(29,36)(30,31)(32,33)(34,39)(35,40)(37,42)(38,41);(3,4,6,8)(5,7,9,10)(12,16)(13,19)(14,21)(24,26)(28,33)(30,35)(34,36)(37,41);(3,5)(4,7)(6,9)(8,10)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,29,32,39)(28,34,33,36)(30,37,35,41)(31,38,40,42);(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,30,32,35)(28,31,33,40)(29,37,39,41)(34,38,36,42);(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,31)(28,35)(29,38)(30,33)(32,40)(34,41)(36,37)(39,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609627);