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×#36553 |
| 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 | |
6190 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);(3,4)(5,6)(7,10)(8,9)(20,23)(24,26);(4,9)(5,7)(11,12,16,14)(13,18,15,17)(19,20)(21,24)(22,23)(25,26);(3,5,8,7)(4,6,9,10)(11,13,16,15)(12,17,14,18)(19,20)(21,24)(22,23)(25,26);(3,6)(4,5)(7,9)(8,10)(11,14,16,12)(13,17,15,18)(19,21,22,25)(20,24,23,26);(3,7,8,5)(4,10,9,6)(11,15,16,13)(12,18,14,17)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537452);