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×#56526 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2661 | C2 |
| 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);(3,4,8,9)(5,7,6,10)(21,25)(22,29)(28,33)(30,34);(4,9)(5,6)(11,12)(13,17)(14,16)(15,18)(19,20,24,27)(21,22,25,29)(23,31,32,26)(28,34,33,30);(3,5,8,6)(4,10,9,7)(11,12,14,16)(13,15,18,17)(19,21)(20,22)(23,28)(24,25)(26,30)(27,29)(31,34)(32,33);(3,6,8,5)(4,7,9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,25)(21,27)(23,30)(24,29)(26,33)(28,31)(32,34);(3,7)(4,5)(6,9)(8,10)(11,13)(12,15)(14,18)(16,17)(19,23,24,32)(20,26,27,31)(21,28,25,33)(22,30,29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006244);