Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62609 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6212 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)(5,12)(6,10)(7,16)(8,11)(9,14)(13,18)(15,17)(19,20,24,29)(21,31,27,26)(22,33,34,32)(23,30,25,28);(3,5)(4,9)(6,13)(7,15)(8,14)(10,17)(11,12)(16,18)(19,21)(20,26)(22,30)(23,33)(24,27)(25,32)(28,34)(29,31);(3,6,8,7)(4,10,11,16)(5,13,14,15)(9,17,12,18)(19,22,23,26,24,34,25,31)(20,27,28,32,29,21,30,33);(5,14)(9,12)(13,15)(17,18)(21,27)(22,34)(26,31)(32,33);(3,7,8,6)(4,10,11,16)(5,15,14,13)(9,17,12,18)(19,23,24,25)(20,28,29,30)(21,32,27,33)(22,31,34,26)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10487225);