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×#48158 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5240 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);(11,12,15,16)(13,14,17,18)(19,20)(21,22)(23,31)(24,25)(26,29)(27,32)(28,33)(30,34);(3,4)(5,9)(6,10)(7,8)(11,12,15,16)(13,18,17,14)(20,25)(22,26)(27,31)(30,33);(4,8)(5,10)(12,16)(13,17)(19,21)(20,26)(22,25)(23,28)(24,29)(27,33)(30,31)(32,34);(3,5)(4,6)(7,10)(8,9)(11,13,15,17)(12,14,16,18)(19,22,24,26)(20,21,25,29)(23,30,32,33)(27,28,31,34);(3,6)(4,5)(7,9)(8,10)(11,14)(12,17)(13,16)(15,18)(19,23,24,32)(20,27,25,31)(21,28,29,34)(22,30,26,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610967);