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×#41695 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2813 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5212 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);(4,7)(8,10)(11,12)(13,17)(14,16)(15,18)(19,20,24,28)(21,23,27,25)(22,26,32,34)(29,31,30,33);(11,12,14,16)(13,15,18,17)(21,27)(23,25)(29,30)(31,33);(3,4,6,7)(5,8,9,10)(11,12)(13,15)(14,16)(17,18)(19,21,24,27)(20,25,28,23)(22,30,32,29)(26,31,34,33);(11,13,14,18)(12,15,16,17)(19,22)(20,26)(21,29)(23,31)(24,32)(25,33)(27,30)(28,34);(3,5,6,9)(4,8,7,10)(19,23,24,25)(20,27,28,21)(22,31,32,33)(26,30,34,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547028);