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×#41391 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(13,17)(15,18)(19,20,24,25)(21,27,29,23)(22,30,33,26)(28,32,31,34);(3,4,6,8)(5,7,10,9)(11,12,14,16)(13,15,17,18)(20,25)(23,27)(26,30)(32,34);(3,4)(5,7)(6,8)(9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,23)(22,31)(24,29)(25,27)(26,34)(28,33)(30,32);(3,5,6,10)(4,7,8,9)(19,22,24,33)(20,26,25,30)(21,28,29,31)(23,32,27,34);(19,23)(20,21)(22,32)(24,27)(25,29)(26,28)(30,31)(33,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547025);