Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41518 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5094 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)(5,6,10,9)(11,12,15,18)(13,16,14,17)(19,20)(21,23)(22,30)(24,25)(26,28)(27,32)(29,33)(31,34);(4,8)(6,9)(11,13,15,14)(12,16,18,17)(20,25)(21,28)(27,30)(29,34);(4,8)(5,10)(11,12)(13,16)(14,17)(15,18)(19,21)(20,26)(22,29)(23,25)(24,28)(27,33)(30,31)(32,34);(3,5,7,10)(4,9,8,6)(11,14,15,13)(12,17,18,16)(19,22,24,32)(20,27,25,30)(21,29,28,34)(23,31,26,33);(3,6,7,9)(4,10,8,5)(19,23,24,26)(20,21,25,28)(22,31,32,33)(27,29,30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547176);