Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#50582 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,6)(7,8)(9,10)(12,16)(17,18)(19,20,24,25)(21,22,26,30)(23,31,32,27)(28,34,33,29);(4,8)(6,9)(20,25)(21,26)(23,32)(29,34);(3,5)(4,9)(6,8)(7,10)(11,12,15,16)(13,17,14,18)(19,21)(20,22)(23,28)(24,26)(25,30)(27,29)(31,34)(32,33);(3,6,7,9)(4,5,8,10)(11,13,15,14)(12,17,16,18)(19,22)(20,26)(21,25)(23,29)(24,30)(27,33)(28,31)(32,34);(3,5)(4,9)(6,8)(7,10)(11,14,15,13)(12,18,16,17)(19,23,24,32)(20,27,25,31)(21,28,26,33)(22,29,30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613164);