Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37177 |
| 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 | |
4546 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);(4,7)(8,10)(14,21)(18,23)(20,25)(24,26)(28,32)(29,35)(34,39)(36,41);(12,16)(13,19)(18,23)(20,25)(27,28,31,32)(29,33,35,38)(30,34,37,39)(36,40,41,42);(3,4,6,7)(5,8,9,10)(11,12)(13,17)(14,18)(15,16)(19,22)(20,24)(21,23)(25,26)(27,28)(29,33)(30,34)(31,32)(35,38)(36,40)(37,39)(41,42);(11,13,15,19)(12,17,16,22)(14,20,21,25)(18,24,23,26)(27,29)(28,33)(30,36)(31,35)(32,38)(34,40)(37,41)(39,42);(3,5,6,9)(4,8,7,10)(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,30,31,37)(28,34,32,39)(29,36,35,41)(33,40,38,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541668);