Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#33281 |
| 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 | |
1842 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)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18)(19,20,23,26)(21,27,28,34)(22,30,31,25)(24,32,33,29);(4,8)(9,10)(11,13,14,17)(12,15,16,18)(21,28)(24,33)(27,34)(29,32);(3,5,7,6)(4,9,8,10)(19,21,22,29,23,28,31,32)(20,24,25,27,26,33,30,34);(3,6,7,5)(4,10,8,9)(12,16)(15,18)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537867);