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×#41759 |
| 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 | |
4898 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);(3,4,7,6)(5,9,10,8)(12,14)(16,17)(19,20)(21,23)(22,30)(24,25)(26,29)(27,33)(28,34)(31,32);(3,4,7,6)(5,8,10,9)(11,12,15,14)(13,17,18,16)(20,25)(23,26)(27,30)(32,34);(3,5,7,10)(4,8,6,9)(11,13,15,18)(12,16,14,17)(19,21)(20,26)(22,31)(23,25)(24,29)(27,32)(28,33)(30,34);(11,14,15,12)(13,17,18,16)(19,22,24,33)(20,27,25,30)(21,28,29,31)(23,32,26,34);(3,6,7,4)(5,9,10,8)(19,23,24,26)(20,21,25,29)(22,32,33,34)(27,28,30,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547073);