Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#47201 |
| 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 | |
5022 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);(3,4,7,5)(6,9,10,8)(13,17)(15,18)(19,20,24,26)(21,27,28,23)(22,25,32,34)(29,33,30,31);(3,4)(5,7)(6,9)(8,10)(11,12,14,16)(13,15,17,18)(21,28)(23,27)(29,30)(31,33);(4,5)(8,9)(11,13)(12,15)(14,17)(16,18)(19,21)(20,23)(22,30)(24,28)(25,33)(26,27)(29,32)(31,34);(3,5,7,4)(6,9,10,8)(19,22)(20,25)(21,29)(23,31)(24,32)(26,34)(27,33)(28,30);(3,6,7,10)(4,8,5,9)(19,23)(20,21)(22,31)(24,27)(25,29)(26,28)(30,34)(32,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608637);