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×#45797 |
| 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 | |
4838 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)(5,9,10,6)(11,12,14,16)(13,17,18,15)(19,20)(21,27)(22,30)(23,26)(24,29)(25,31)(28,32)(33,34);(5,10)(6,9)(19,20,23,26)(21,24,29,27)(22,30,31,25)(28,33,34,32);(3,5)(4,6)(7,10)(8,9)(11,13,14,18)(12,15,16,17)(19,20)(21,24)(22,25)(23,26)(27,29)(28,32)(30,31)(33,34);(3,5)(4,6)(7,10)(8,9)(19,21,23,29)(20,24,26,27)(22,28,31,34)(25,32,30,33);(3,6)(4,5)(7,9)(8,10)(19,22)(20,25)(21,28)(23,31)(24,32)(26,30)(27,33)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609626);