Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36043 |
| 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 | |
5458 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);(4,5)(8,9)(11,12,15,18)(13,16,14,17)(19,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,34)(32,33);(3,4,7,5)(6,8,10,9)(11,12)(13,16)(14,17)(15,18)(20,25)(21,27)(28,33)(29,34);(3,5,7,4)(6,9,10,8)(11,13,15,14)(12,16,18,17)(19,21,24,27)(20,26,25,22)(23,29,32,34)(28,31,33,30);(3,6,7,10)(4,8,5,9)(11,14,15,13)(12,17,18,16)(19,22,24,26)(20,27,25,21)(23,31,32,30)(28,29,33,34);(19,23)(20,28)(21,29)(22,30)(24,32)(25,33)(26,31)(27,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536664);