Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#49315 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5120 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);(3,4,7,8)(5,9,10,6)(13,17)(16,18)(19,20)(21,28)(22,26)(23,31)(24,25)(27,32)(29,30)(33,34);(4,8)(6,9)(11,12,15,14)(13,16,17,18)(20,25)(21,26)(23,32)(30,34);(3,5)(4,6)(7,10)(8,9)(11,13,15,17)(12,16,14,18)(19,21)(20,22)(23,29)(24,26)(25,28)(27,30)(31,34)(32,33);(3,6)(4,5)(7,9)(8,10)(19,22)(20,26)(21,25)(23,30)(24,28)(27,33)(29,31)(32,34);(3,5)(4,6)(7,10)(8,9)(11,14,15,12)(13,18,17,16)(19,23)(20,27)(21,29)(22,30)(24,32)(25,31)(26,33)(28,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7611932);