Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46309 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4480 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);(4,8)(9,10)(13,19)(17,22)(20,25)(24,26)(27,28,31,35)(29,33,37,32)(30,38,39,34)(36,40,42,41);(12,16)(13,19)(14,21)(24,26)(27,29,31,37)(28,32,35,33)(30,36,39,42)(34,40,38,41);(3,4,7,8)(5,9,6,10)(11,12)(13,17)(14,18)(15,16)(19,22)(20,24)(21,23)(25,26)(27,29)(28,33)(30,36)(31,37)(32,35)(34,41)(38,40)(39,42);(3,5,7,6)(4,9,8,10)(11,13,15,19)(12,17,16,22)(14,20,21,25)(18,24,23,26)(27,30,31,39)(28,34,35,38)(29,36,37,42)(32,40,33,41);(3,6,7,5)(4,10,8,9)(11,14,15,21)(12,18,16,23)(13,20,19,25)(17,24,22,26)(27,28)(29,33)(30,34)(31,35)(32,37)(36,41)(38,39)(40,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610189);