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×#36418 |
| 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 | |
5660 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);(13,17)(16,18)(20,25)(21,26);(3,4,6,8)(5,7,10,9)(11,12,15,14)(13,16,17,18)(19,20)(21,22)(23,26)(24,25);(3,4)(5,9)(6,8)(7,10)(11,13)(12,16)(14,18)(15,17)(19,21,24,26)(20,22,25,23);(3,5)(4,7)(6,10)(8,9)(11,12,15,14)(13,16,17,18)(19,22,24,23)(20,21,25,26);(3,4)(5,9)(6,8)(7,10)(11,14,15,12)(13,18,17,16)(19,23,24,22)(20,26,25,21)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537275);