Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52608 |
| 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 | |
4598 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)(13,18)(14,22)(15,20)(16,17)(19,23)(21,24)(25,26)(27,28,32,30)(29,35,37,33)(31,38,39,34)(36,40,42,41);(3,4)(5,8)(6,7)(9,10)(12,17)(13,20)(14,23)(24,25)(27,29,32,37)(28,33,30,35)(31,36,39,42)(34,40,38,41);(4,7)(8,10)(11,13,16,20)(12,18,17,15)(14,21,23,26)(19,25,22,24)(27,29)(28,33)(30,35)(31,36)(32,37)(34,40)(38,41)(39,42);(3,4)(5,8)(6,7)(9,10)(11,14)(12,19)(13,21)(15,24)(16,23)(17,22)(18,25)(20,26)(27,30,32,28)(29,33,37,35)(31,38,39,34)(36,40,42,41);(3,5)(4,8)(6,9)(7,10)(11,15,16,18)(12,20,17,13)(14,24,23,25)(19,26,22,21)(27,31,32,39)(28,34,30,38)(29,36,37,42)(33,40,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999552);