Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46144 |
| 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 | |
4930 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)(6,10)(7,8)(13,17)(15,18)(19,20,24,22)(21,25,29,27)(23,30,31,26)(28,33,34,32);(4,7)(5,10)(11,12,14,16)(13,15,17,18)(20,22)(25,27)(26,30)(32,33);(3,5,8,10)(4,9,7,6)(11,13)(12,15)(14,17)(16,18)(19,21,24,29)(20,25,22,27)(23,28,31,34)(26,32,30,33);(3,6)(4,5)(7,10)(8,9)(19,22,24,20)(21,27,29,25)(23,30,31,26)(28,33,34,32);(3,7,8,4)(5,6,10,9)(19,23,24,31)(20,26,22,30)(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, 7610007);