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×#52933 |
| 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 | |
4636 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);(3,4,6,8)(5,9,10,7)(14,22)(18,23)(20,26)(24,25)(27,28,32,29)(30,35,37,33)(31,34,39,36)(38,41,42,40);(5,10)(7,9)(12,16)(17,19)(18,23)(24,25)(28,29)(30,37)(34,36)(38,42);(3,5)(4,7)(6,10)(8,9)(11,12,15,16)(13,19,21,17)(14,18,22,23)(20,25,26,24)(27,29,32,28)(30,35,37,33)(31,36,39,34)(38,41,42,40);(3,4)(5,7)(6,8)(9,10)(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25)(27,30)(28,33)(29,35)(31,38)(32,37)(34,40)(36,41)(39,42);(11,14)(12,18)(13,20)(15,22)(16,23)(17,24)(19,25)(21,26)(27,31,32,39)(28,34,29,36)(30,38,37,42)(33,40,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999947);