Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#55736 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2424 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,17)(16,18)(19,20,22,24)(21,25,26,23);(6,10)(8,9)(12,13)(16,17)(19,20,22,24)(21,23,26,25);(4,5)(8,9)(11,12,15,13)(14,16,18,17)(19,20)(21,23)(22,24)(25,26);(3,5,7,4)(6,9,10,8)(11,13,15,12)(14,17,18,16)(19,21,22,26)(20,23,24,25);(3,6)(4,8)(5,9)(7,10)(11,14)(12,16)(13,17)(15,18)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006049);