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×#50569 |
| 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 | |
4948 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);(3,4,8,6)(5,7,10,9)(11,12,15,18)(13,16,14,17)(19,20,24,27)(21,23,26,32)(22,25,29,33)(28,31,30,34);(3,5)(4,9)(6,7)(8,10)(11,13,15,14)(12,16,18,17)(21,26)(22,29)(23,32)(25,33);(4,6)(5,10)(11,12)(13,16)(14,17)(15,18)(19,21,24,26)(20,23,27,32)(22,30,29,28)(25,34,33,31);(3,6,8,4)(5,7,10,9)(11,13,15,14)(12,16,18,17)(19,22)(20,25)(21,28)(23,31)(24,29)(26,30)(27,33)(32,34);(3,7)(4,5)(6,10)(8,9)(11,14,15,13)(12,17,18,16)(19,23)(20,26)(21,27)(22,31)(24,32)(25,30)(28,33)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613260);