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×#52611 |
| 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 | |
4674 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)(13,14)(15,19)(16,17)(18,21)(20,22)(23,24)(25,26)(27,28,33,36)(29,34,30,31)(32,40,41,35)(37,39,38,42);(3,4)(5,8)(6,7)(9,10)(12,17)(14,18)(19,24)(22,25)(27,29,33,30)(28,34,36,31)(32,37,41,38)(35,42,40,39);(4,7)(8,10)(11,13)(12,18)(14,17)(15,20)(16,21)(19,25)(22,24)(23,26)(27,30,33,29)(28,34,36,31)(32,38,41,37)(35,42,40,39);(3,4)(5,8)(6,7)(9,10)(11,14,16,18)(12,13,17,21)(15,22,23,25)(19,20,24,26)(27,31,33,34)(28,30,36,29)(32,39,41,42)(35,38,40,37);(3,5)(4,8)(6,9)(7,10)(11,15,16,23)(12,19,17,24)(13,20,21,26)(14,22,18,25)(27,32)(28,35)(29,37)(30,38)(31,39)(33,41)(34,42)(36,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999582);