Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#17234 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,7)(8,10)(11,12,14,16)(13,15,17,18)(19,20,22,23)(21,24,26,25)(28,29)(32,33);(4,7)(5,9)(13,17)(15,18)(20,23)(24,25)(27,28,31,29)(30,32,34,33);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21,22,26)(20,24,23,25)(27,29,31,28)(30,32,34,33);(3,5)(4,8)(6,9)(7,10)(11,13)(12,15)(14,17)(16,18)(19,20)(21,25)(22,23)(24,26)(27,30)(28,32)(29,33)(31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752276);