Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#40014 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4,7,9)(5,6,10,8)(19,20,23,21)(22,25,26,24);(3,5,7,10)(4,8,9,6)(12,16)(14,17)(19,21,23,20)(22,25,26,24);(3,6)(4,5)(7,8)(9,10)(11,12)(13,17)(14,18)(15,16)(19,22,23,26)(20,24,21,25);(3,6)(4,5)(7,8)(9,10)(11,13,15,18)(12,17,16,14);(3,5)(4,6)(7,10)(8,9)(11,14,15,17)(12,18,16,13)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544722);