Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#39769 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| 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) = 512 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(4,7)(8,10)(11,12,16,13)(14,17,15,18)(20,24)(21,27)(26,29)(28,33);(4,7)(5,9)(11,13,16,12)(14,17,15,18)(19,20)(21,25)(22,29)(23,24)(26,30)(27,31)(28,34)(32,33);(3,4)(5,8)(6,7)(9,10)(11,14,16,15)(12,17,13,18);(19,21,23,27)(20,25,24,31)(22,28,30,33)(26,32,29,34);(3,5,6,9)(4,8,7,10)(11,15,16,14)(12,18,13,17)(19,22,23,30)(20,26,24,29)(21,28,27,33)(25,32,31,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544621);