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×#41602 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2825 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5180 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);(4,6)(9,10)(11,12,15,14)(13,17,18,16)(19,20)(21,27)(22,28)(23,26)(24,25)(29,33)(30,34)(31,32);(3,4,8,6)(5,9,7,10)(11,12,15,14)(13,16,18,17)(20,25)(21,28)(26,32)(29,34);(3,5,8,7)(4,9,6,10)(11,13,15,18)(12,16,14,17)(19,21)(20,22)(23,29)(24,28)(25,27)(26,30)(31,34)(32,33);(3,6,8,4)(5,9,7,10)(11,14,15,12)(13,17,18,16)(19,22)(20,21)(23,30)(24,27)(25,28)(26,29)(31,33)(32,34);(3,7,8,5)(4,9,6,10)(19,23,24,31)(20,26,25,32)(21,29,28,34)(22,30,27,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547387);