Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46399 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2812 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5882 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);(11,12,15,17)(13,18,16,14)(19,20,23,21)(22,25,26,24);(3,4)(5,9)(6,7)(8,10)(11,13,15,16)(12,14,17,18)(19,21,23,20)(22,25,26,24);(4,7)(8,9)(11,14,15,18)(12,16,17,13)(19,22)(20,24)(21,25)(23,26);(3,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,18)(16,17);(3,5)(4,8)(6,10)(7,9)(11,13)(12,14)(15,16)(17,18)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7607337);