Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508022 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1175 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1175 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,19)(13,23)(14,18)(16,20)(17,21)(22,24);(2,5)(6,8)(9,10)(11,19)(12,15)(13,23)(14,18)(16,22)(17,21)(20,24);(1,3)(2,6)(4,7)(5,8)(10,15)(11,20)(12,22)(13,21)(16,17)(18,19)(23,24);(3,7)(6,8)Polycyclic presentation:
2^9|f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,d^c=de,f^c=fh,g^c=gh,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415248);