Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13427 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,5)(7,8,10,12)(9,15,16,22)(11,19,18,14)(13,20,21,17);(3,4)(5,6)(8,12)(9,13)(11,18)(15,17)(16,21)(20,22);(3,5,6,4)(7,9,10,16)(8,13,12,21)(11,20,18,17)(14,22,19,15)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406301);