Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#29349 |
| 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 | |
1894 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)(6,7)(8,10)(13,17)(15,18)(19,20,24,26,23,28,22,27)(21,25,31,32,30,34,29,33);(4,7)(8,9)(11,12,14,16)(13,18,17,15)(19,21,23,30)(20,25,28,34)(22,29,24,31)(26,32,27,33);(11,13,14,17)(12,15,16,18)(20,26)(22,24)(25,32)(27,28)(29,31)(33,34);(3,5,6,10)(4,8,7,9)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371309);