Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26850 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,26)(24,25);(3,5,7,6)(4,8,9,10)(11,13,17,20,18,24,16,23)(12,19,14,25,22,26,21,15);(3,6,7,5)(4,8,9,10)(11,14,16,12,18,21,17,22)(13,19,23,15,24,26,20,25);(4,9)(8,10)(11,15,17,19,18,25,16,26)(12,20,14,24,22,23,21,13)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351152);