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×#47763 |
| 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 | |
5114 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)(5,10)(6,9)(7,8)(11,12,15,13)(14,17,18,16)(19,20,24,22)(21,25,27,29)(23,26,31,30)(28,32,33,34);(3,5)(4,9)(6,8)(7,10)(11,13,15,12)(14,17,18,16)(21,27)(23,31)(25,29)(26,30);(5,6)(9,10)(11,14)(12,16)(13,17)(15,18)(19,21)(20,25)(22,29)(23,28)(24,27)(26,32)(30,34)(31,33);(3,6,8,5)(4,9,7,10)(19,22,24,20)(21,25,27,29)(23,30,31,26)(28,32,33,34);(3,7,8,4)(5,9,6,10)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608940);