Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501040 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1379 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)(9,10,11,15,19,23,24,22,13,17,20,21,12,16,18,14);(2,4)(5,6)(7,8);(6,8)(10,14)(11,18)(12,19)(15,16)(17,22)(20,24)(21,23);(2,4)Polycyclic presentation:
2^9|a^2=f,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,h^c=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72310027, D4×#2184, #72310184 |
| maximal normal subgroup of | #72310027, D4×#2184, #72310184 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408236);