Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#45308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,7,6)(5,9,10,8)(11,12,15,16)(13,17,18,14)(22,30)(25,31)(28,34)(32,33);(12,16)(14,17)(19,20,23,26)(21,24,29,27)(22,31,30,25)(28,33,34,32);(3,5,7,10)(4,8,6,9)(11,13)(12,14)(15,18)(16,17)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,31)(32,34);(3,6,7,4)(5,9,10,8)(11,14)(12,13)(15,17)(16,18)(19,21)(20,24)(22,28)(23,29)(25,32)(26,27)(30,34)(31,33);(19,22,23,30)(20,25,26,31)(21,28,29,34)(24,32,27,33)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7577189);