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×#54713 |
| 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 | |
4664 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)(5,7)(6,9)(8,10)(12,16)(13,19)(18,23)(21,25)(27,28,32,33)(29,30,37,34)(31,39,40,35)(36,41,42,38);(4,5)(8,9)(14,22)(18,23)(21,25)(24,26)(28,33)(30,34)(31,40)(36,42);(3,4,7,5)(6,8,10,9)(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24)(27,29,32,37)(28,34,33,30)(31,36,40,42)(35,41,39,38);(3,5,7,4)(6,9,10,8)(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,30,32,34)(28,29,33,37)(31,38,40,41)(35,36,39,42);(3,6)(4,8)(5,9)(7,10)(11,14,15,22)(12,18,16,23)(13,21,19,25)(17,24,20,26)(27,31)(28,35)(29,36)(30,38)(32,40)(33,39)(34,41)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006200);