Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#52988 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7770 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);(5,9)(8,10)(12,15)(13,17)(19,20)(21,25)(22,24)(23,26);(3,4)(5,8)(6,7)(9,10)(20,22)(25,26);(4,7)(8,10)(11,12,14,15)(13,16,17,18)(19,21,24,23)(20,25,22,26);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,22,24,20)(21,25,23,26);(3,5,6,9)(4,8,7,10)(11,13,14,17)(12,16,15,18)(19,23,24,21)(20,25,22,26)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000259);