Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#29209 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,10)(5,9,6,8)(11,12,14,16)(13,15,17,18)(19,20)(21,27)(22,24)(23,31)(25,29)(26,30)(28,33)(32,34);(3,5,7,6)(4,8,10,9)(11,13,14,17)(12,15,16,18)(20,24)(21,28)(26,31)(29,34);(13,17)(15,18)(19,21,23,29,22,28,30,34)(20,25,26,27,24,32,31,33);(3,6,7,5)(4,9,10,8)(20,24)(25,32)(26,31)(27,33)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375275);