Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#27591 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,14,16)(13,15,17,18)(19,20)(21,24)(22,25)(23,27)(26,30)(28,32)(29,31)(33,34);(3,5,6,10)(4,7,8,9)(11,13)(12,15)(14,17)(16,18)(19,21)(20,24)(22,29)(23,28)(25,31)(26,33)(27,32)(30,34);(13,17)(15,18)(20,25)(22,30)(24,31)(26,27)(29,34)(32,33);(4,8)(7,9)(21,28)(24,32)(29,34)(31,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365989);