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×#39576 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2688 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5684 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);(4,8)(9,10)(11,12,15,14)(13,16,18,17)(19,20,22,23)(21,25,26,24);(11,13)(12,16)(14,17)(15,18)(20,23)(24,25);(3,4)(5,9)(6,10)(7,8)(11,12,15,14)(13,17,18,16)(19,20)(21,24)(22,23)(25,26);(3,5,7,6)(4,9,8,10)(19,21)(20,24)(22,26)(23,25);(3,6,7,5)(4,10,8,9)(11,14,15,12)(13,16,18,17)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544022);