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×#53921 |
| 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 | |
4652 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);(3,4,6,8)(5,9,10,7)(12,16)(13,19)(18,23)(21,25)(27,28,33,31)(29,36,30,34)(32,39,40,35)(37,41,38,42);(5,10)(7,9)(14,22)(18,23)(21,25)(24,26)(27,29,33,30)(28,34,31,36)(32,37,40,38)(35,41,39,42);(3,5)(4,7)(6,10)(8,9)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,30,33,29)(28,34,31,36)(32,38,40,37)(35,41,39,42);(3,4)(5,7)(6,8)(9,10)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,25)(23,26)(27,31,33,28)(29,34,30,36)(32,39,40,35)(37,41,38,42);(11,14)(12,18)(13,21)(15,22)(16,23)(17,24)(19,25)(20,26)(27,32)(28,35)(29,37)(30,38)(31,39)(33,40)(34,41)(36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009980);