Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#48923 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2424 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4,7,9)(5,10,6,8)(11,12,16,18)(13,17,14,15)(20,23)(24,26);(3,5,7,6)(4,8,9,10)(11,13)(12,17)(14,16)(15,18);(5,6)(8,10)(12,18)(13,14)(19,20)(21,24)(22,23)(25,26);(3,6,7,5)(4,8,9,10)(11,14,16,13)(12,15,18,17)(19,21,22,25)(20,24,23,26);(3,4)(5,10)(6,8)(7,9)(11,15)(12,13)(14,18)(16,17)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7611358);