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×#36566 |
| 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 | |
6018 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)(5,8)(6,9)(7,10)(20,23)(24,26);(3,5,6,10)(4,7,9,8)(11,12,16,14)(13,18,15,17)(19,20)(21,24)(22,23)(25,26);(3,4,6,9)(5,8,10,7)(11,13,16,15)(12,17,14,18);(3,4)(5,7)(6,9)(8,10)(11,14,16,12)(13,17,15,18);(3,5)(4,8)(6,10)(7,9)(11,15,16,13)(12,18,14,17)(19,21,22,25)(20,24,23,26)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537468);