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×#36589 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5656 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);(3,4,8,6)(5,9,7,10)(12,13)(16,17)(20,24)(25,26);(3,4,8,6)(5,10,7,9)(11,12,15,13)(14,16,18,17)(19,20)(21,25)(22,26)(23,24);(3,5,8,7)(4,9,6,10)(11,13,15,12)(14,17,18,16)(19,21,23,22)(20,25,24,26);(3,6,8,4)(5,9,7,10)(11,14,15,18)(12,16,13,17)(19,22,23,21)(20,26,24,25);(3,7,8,5)(4,10,6,9)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537496);