Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#16120 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3130 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);(4,7)(5,9)(11,12)(13,17)(14,15)(16,18)(19,20,22,24)(21,25,26,23)(27,28)(29,31)(30,32)(33,34);(12,15)(13,18)(21,26)(23,25)(27,28,30,32)(29,33,34,31);(3,4,6,7)(5,8,9,10)(11,12)(13,16)(14,15)(17,18)(19,21,22,26)(20,23,24,25)(27,28)(29,31)(30,32)(33,34);(3,5)(4,8)(6,9)(7,10)(11,13,14,18)(12,16,15,17)(19,20)(21,23)(22,24)(25,26)(27,29,30,34)(28,31,32,33)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 517409);