Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#23785 |
| 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 | |
1870 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)(5,8)(6,7)(9,10)(11,12,14,16)(13,17,18,15)(19,20,23,27)(21,29,32,25)(22,26,24,28)(30,34,31,33);(4,7)(8,10)(19,21,23,32)(20,25,27,29)(22,31,24,30)(26,34,28,33);(11,13,14,18)(12,15,16,17)(21,30)(22,24)(25,33)(26,28)(29,34)(31,32);(3,5,6,9)(4,8,7,10)(19,22,23,24)(20,26,27,28)(21,31,32,30)(25,34,29,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339731);