Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#56666 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5000 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);(3,4,8,6)(5,9,7,10)(11,12)(13,16)(14,18)(15,17)(19,20,24,27)(21,23,26,32)(22,25,31,34)(28,30,29,33);(4,6)(5,7)(11,13,15,18)(12,16,17,14)(21,26)(23,32)(28,29)(30,33);(3,5,8,7)(4,9,6,10)(11,13)(12,14)(15,18)(16,17)(19,21,24,26)(20,23,27,32)(22,29,31,28)(25,33,34,30);(3,6,8,4)(5,9,7,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,25)(21,28)(23,30)(24,31)(26,29)(27,34)(32,33);(3,7,8,5)(4,9,6,10)(19,23)(20,26)(21,27)(22,30)(24,32)(25,29)(28,34)(31,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007019);