Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52885 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(4,9)(5,7)(11,12,15,13)(14,17,18,16)(19,20,23,26)(21,27,28,24)(22,25,30,32)(29,33,34,31);(3,4,8,9)(5,6,7,10)(12,13)(14,18)(21,28)(24,27)(29,34)(31,33);(3,5,8,7)(4,10,9,6)(11,13,15,12)(14,17,18,16)(19,21)(20,24)(22,29)(23,28)(25,31)(26,27)(30,34)(32,33);(3,6,8,10)(4,5,9,7)(11,14,15,18)(12,16,13,17)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,34);(3,7,8,5)(4,10,9,6)(19,22,23,30)(20,25,26,32)(21,29,28,34)(24,31,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999765);