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×#49017 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,10)(6,9)(7,8)(11,12,15,18)(13,16,14,17)(19,20)(21,27)(22,30)(23,24)(25,29)(26,31)(28,32)(33,34);(3,5,8,6)(4,9,7,10)(11,13,15,14)(12,16,18,17)(20,24)(25,27)(26,30)(32,33);(5,6)(9,10)(11,12)(13,16)(14,17)(15,18)(19,21)(20,25)(22,28)(23,29)(24,27)(26,32)(30,33)(31,34);(3,6,8,5)(4,9,7,10)(11,14,15,13)(12,17,18,16)(19,20)(21,25)(22,26)(23,24)(27,29)(28,32)(30,31)(33,34);(3,7,8,4)(5,9,6,10)(19,22,23,31)(20,26,24,30)(21,28,29,34)(25,32,27,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612084);