Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#49187 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5050 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);(3,4,7,9)(5,8,10,6)(13,17)(16,18)(19,20)(21,22)(23,31)(24,25)(26,32)(27,30)(28,34)(29,33);(3,5,7,10)(4,8,9,6)(11,12,15,14)(13,16,17,18)(20,25)(21,27)(23,32)(29,34);(4,9)(5,10)(11,13)(12,16)(14,18)(15,17)(19,21,24,27)(20,22,25,30)(23,28,32,33)(26,29,31,34);(3,5)(4,8)(6,9)(7,10)(19,22)(20,21)(23,29)(24,30)(25,27)(26,28)(31,33)(32,34);(3,6,7,8)(4,5,9,10)(11,14,15,12)(13,18,17,16)(19,23)(20,26)(21,28)(22,29)(24,32)(25,31)(27,33)(30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7611421);