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×#51482 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4854 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)(5,9)(6,7)(8,10)(11,12,15,13)(14,17,18,16)(19,20,24,25)(21,29,30,26)(22,27,28,23)(31,34,33,32);(4,7)(8,9)(11,13,15,12)(14,17,18,16)(20,25)(21,30)(23,27)(31,33);(11,14)(12,16)(13,17)(15,18)(19,21,24,30)(20,26,25,29)(22,33,28,31)(23,32,27,34);(3,4)(5,8)(6,7)(9,10)(19,22,24,28)(20,27,25,23)(21,31,30,33)(26,32,29,34);(3,5,6,10)(4,8,7,9)(19,23,24,27)(20,28,25,22)(21,32,30,34)(26,31,29,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614536);