Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#38255 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5734 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);(12,15)(16,18)(19,20,25,27)(21,28,22,24)(23,31,33,26)(29,32,30,34);(4,7)(5,9)(12,15)(13,17)(19,21,25,22)(20,24,27,28)(23,29,33,30)(26,32,31,34);(3,4,6,7)(5,8,9,10)(11,12)(13,16)(14,15)(17,18)(19,22,25,21)(20,24,27,28)(23,30,33,29)(26,32,31,34);(19,23,25,33)(20,26,27,31)(21,29,22,30)(24,32,28,34);(3,5)(4,8)(6,9)(7,10)(11,13,14,17)(12,16,15,18)(19,24)(20,21)(22,27)(23,32)(25,28)(26,29)(30,31)(33,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542808);