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×#47032 |
| 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 | |
5706 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)(13,15)(14,16)(17,18)(19,20,24,25)(21,29,30,26)(22,27,32,31)(23,33,34,28);(3,4,7,8)(5,6,9,10)(11,12)(13,17)(14,16)(15,18)(20,25)(21,30)(22,32)(23,34);(4,8)(5,9)(11,12,14,16)(13,15,18,17)(19,21,24,30)(20,26,25,29)(22,33,32,28)(23,31,34,27);(3,5)(4,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,27)(21,28)(23,29)(24,32)(25,31)(26,34)(30,33);(3,6)(4,5)(7,10)(8,9)(11,13)(12,15)(14,18)(16,17)(19,23,24,34)(20,28,25,33)(21,31,30,27)(22,29,32,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608360);