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×#57745 |
| 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 | |
4712 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)(5,9)(6,7)(8,10)(12,16)(13,19)(18,23)(21,25)(27,28,32,35)(29,30,37,33)(31,39,40,34)(36,41,42,38);(4,7)(8,9)(14,22)(18,23)(21,25)(24,26)(27,29,32,37)(28,33,35,30)(31,36,40,42)(34,41,39,38);(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,30,32,33)(28,29,35,37)(31,38,40,41)(34,36,39,42);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,25)(23,26)(27,29)(28,33)(30,35)(31,36)(32,37)(34,41)(38,39)(40,42);(3,5,6,10)(4,8,7,9)(11,14,15,22)(12,18,16,23)(13,21,19,25)(17,24,20,26)(27,31)(28,34)(29,36)(30,38)(32,40)(33,41)(35,39)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008623);