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×#57637 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4680 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)(5,8)(6,7)(9,10)(11,12)(13,17)(14,22)(15,16)(18,23)(19,20)(21,26)(24,25)(27,28,32,33)(29,30,37,34)(31,39,40,35)(36,41,42,38);(3,5,7,10)(4,8,6,9)(12,16)(13,19)(14,23)(24,26)(28,33)(30,34)(31,40)(36,42);(4,6)(8,9)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,29,32,37)(28,34,33,30)(31,36,40,42)(35,41,39,38);(3,6,7,4)(5,9,10,8)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,26)(23,25)(27,30,32,34)(28,29,33,37)(31,38,40,41)(35,36,39,42);(3,5)(4,8)(6,9)(7,10)(11,14,15,23)(12,18,16,22)(13,21,19,25)(17,24,20,26)(27,31,32,40)(28,35,33,39)(29,36,37,42)(30,38,34,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008825);