Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#56879 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2464 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5098 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);(3,4,7,5)(6,9,10,8)(11,12,15,17)(13,14,18,16)(19,20)(21,25)(22,26)(23,24)(27,29)(28,33)(30,32)(31,34);(4,5)(8,9)(11,13,15,18)(12,16,17,14)(20,23)(21,27)(22,30)(33,34);(12,17)(14,16)(19,21,24,27)(20,25,23,29)(22,31,30,28)(26,34,32,33);(3,5,7,4)(6,9,10,8)(11,14,15,16)(12,13,17,18)(19,22,24,30)(20,26,23,32)(21,28,27,31)(25,33,29,34);(3,6,7,10)(4,8,5,9)(11,13)(12,16)(14,17)(15,18)(19,23,24,20)(21,29,27,25)(22,32,30,26)(28,34,31,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006854);