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×#49215 |
| 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 | |
5066 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)(5,9)(6,8)(7,10)(11,12,15,18)(13,16,14,17)(19,20)(21,22)(23,31)(24,25)(26,32)(27,30)(28,34)(29,33);(3,4,7,10)(5,8,6,9)(11,13,15,14)(12,16,18,17)(20,25)(21,27)(23,32)(29,34);(3,5,7,6)(4,8,10,9)(11,12)(13,16)(14,17)(15,18)(19,21,24,27)(20,22,25,30)(23,28,32,33)(26,29,31,34);(3,4)(5,8)(6,9)(7,10)(19,22)(20,21)(23,29)(24,30)(25,27)(26,28)(31,33)(32,34);(3,6,7,5)(4,9,10,8)(11,14,15,13)(12,17,18,16)(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, 7611422);