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×#50158 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4920 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);(11,12,15,14)(13,17,18,16)(19,20)(21,28)(22,23)(24,25)(26,29)(27,33)(30,31)(32,34);(4,7)(5,9)(11,12,15,14)(13,16,18,17)(20,25)(21,29)(23,33)(30,32);(3,4,6,7)(5,8,9,10)(11,13,15,18)(12,16,14,17)(19,21)(20,26)(22,32)(23,31)(24,29)(25,28)(27,30)(33,34);(3,4)(5,8)(6,7)(9,10)(11,12,15,14)(13,16,18,17)(19,22,24,27)(20,23,25,33)(21,30,29,32)(26,34,28,31);(3,5)(4,8)(6,9)(7,10)(11,14,15,12)(13,17,18,16)(19,23)(20,27)(21,31)(22,25)(24,33)(26,30)(28,32)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612636);