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×#56524 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5994 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)(5,9)(6,7)(8,10)(11,12)(13,17)(14,18)(15,16);(3,5,7,10)(4,8,6,9)(12,16)(13,18)(19,20)(21,25)(22,24)(23,26);(5,10)(8,9)(11,13)(12,14)(15,18)(16,17)(19,20,22,24)(21,23,26,25);(3,5)(4,9)(6,8)(7,10)(11,13)(12,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,26);(3,6,7,4)(5,8,10,9)(11,14)(12,13)(15,17)(16,18)(19,21)(20,23)(22,26)(24,25)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006065);