Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501046 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#466 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,14,21,20,24,19,15,13,16,11,17,12,18,22,23);(2,4)(5,6)(7,8);(6,8)(10,15)(11,19)(12,20)(14,22)(16,23)(17,18)(21,24);(2,4)(10,16)(15,23)(17,21)(18,24)Polycyclic presentation:
2^9|a^2=fi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72310027, #72310184, #72310196 |
| maximal normal subgroup of | #72310027, #72310184, #72310196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408243);