Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46984 |
| 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 | |
5098 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)(13,17)(16,18)(19,20)(21,29)(22,27)(23,33)(24,25)(26,30)(28,34)(31,32);(4,6)(8,9)(11,12,15,14)(13,16,17,18)(20,25)(21,30)(22,32)(23,34);(3,5)(4,8)(6,9)(7,10)(11,13)(12,16)(14,18)(15,17)(19,21)(20,26)(22,33)(23,31)(24,30)(25,29)(27,34)(28,32);(3,6,7,4)(5,9,10,8)(19,22,24,32)(20,27,25,31)(21,28,30,33)(23,29,34,26);(3,5)(4,8)(6,9)(7,10)(11,14,15,12)(13,18,17,16)(19,23)(20,28)(21,31)(22,29)(24,34)(25,33)(26,32)(27,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7607871);