Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#17236 |
| 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 | |
3098 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)(8,10)(11,12,14,16)(13,15,17,18)(19,20,23,22)(21,24,25,26)(28,29)(30,34);(4,7)(5,9)(13,17)(15,18)(20,22)(24,26)(27,28,31,29)(30,32,34,33);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21,23,25)(20,24,22,26)(27,29,31,28)(30,32,34,33);(3,5)(4,8)(6,9)(7,10)(11,13)(12,15)(14,17)(16,18)(19,22,23,20)(21,24,25,26)(27,30,31,34)(28,32,29,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752278);