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×#36480 |
| 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 | |
5882 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)(13,15)(14,17)(16,18)(19,20,23,26)(21,22,25,24);(3,4)(5,9)(6,10)(7,8)(11,13,16,17)(12,14,18,15)(19,21,23,25)(20,24,26,22);(4,8)(5,10)(11,14,16,15)(12,17,18,13)(19,20)(21,24)(22,25)(23,26);(3,5,7,10)(4,9,8,6)(11,12)(13,14)(15,17)(16,18)(19,22,23,24)(20,25,26,21);(3,6,7,9)(4,5,8,10)(11,15,16,14)(12,13,18,17)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537360);