Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#59782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2555 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5876 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);(4,7)(8,10)(11,12,14,15)(13,17,18,16)(19,20,23,21)(22,24,26,25);(3,4)(5,8)(6,7)(9,10)(12,15)(16,17)(22,26)(24,25);(11,12)(13,16)(14,15)(17,18)(19,21,23,20)(22,25,26,24);(3,5,6,9)(4,8,7,10)(11,12)(13,16)(14,15)(17,18)(19,22)(20,24)(21,25)(23,26);(3,4)(5,8)(6,7)(9,10)(11,13,14,18)(12,16,15,17)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573600);