Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53290 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2671 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5200 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)(5,7)(6,9)(8,10)(13,17)(16,18)(19,20)(21,25)(22,29)(23,26)(24,27)(28,32)(30,34)(31,33);(4,5)(8,9)(12,15)(16,18)(19,21,24,29)(20,25,27,22)(23,28,31,34)(26,32,33,30);(3,4,7,5)(6,8,10,9)(11,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,28)(24,29)(25,27)(26,30)(31,34)(32,33);(3,5,7,4)(6,9,10,8)(11,13)(12,16)(14,17)(15,18)(19,22)(20,21)(23,30)(24,25)(26,28)(27,29)(31,32)(33,34);(3,6,7,10)(4,8,5,9)(11,12)(13,16)(14,15)(17,18)(19,23,24,31)(20,26,27,33)(21,28,29,34)(22,30,25,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012880);