Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#42691 |
| 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 | |
5000 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)(19,20,24,27)(21,25,28,23)(22,26,32,34)(29,33,30,31);(4,5)(8,9)(11,12,14,16)(13,15,17,18)(21,28)(23,25)(29,30)(31,33);(3,5,7,4)(6,9,10,8)(11,13,14,17)(12,15,16,18)(19,21)(20,25)(22,30)(23,27)(24,28)(26,31)(29,32)(33,34);(3,6,7,10)(4,8,5,9)(19,22)(20,26)(21,29)(23,31)(24,32)(25,33)(27,34)(28,30);(19,23,24,25)(20,21,27,28)(22,31,32,33)(26,29,34,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548476);