Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36519 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(11,12,15,14)(13,16,18,17)(21,25)(24,26);(3,4)(5,9)(6,10)(7,8)(12,14)(16,17)(19,20,23,22)(21,24,25,26);(4,8)(5,10)(11,12,15,14)(13,17,18,16)(19,21)(20,24)(22,26)(23,25);(3,5,7,10)(4,9,8,6)(11,13)(12,16)(14,17)(15,18)(19,22,23,20)(21,26,25,24);(3,6,7,9)(4,5,8,10)(11,14,15,12)(13,16,18,17)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537403);