Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144228 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C2×H(5,𝔽2) | D4 |
| LCS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
669 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 669 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 24 points:
(1,2,4,7)(3,5,8,6)(14,23)(18,21);(2,5)(3,8)(6,7)(9,10,13,15)(11,17,19,24)(12,20,22,16)(14,21,23,18);(9,11)(10,14)(12,21)(13,19)(15,23)(16,24)(17,20)(18,22)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,f^c=fi,h^c=hi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51465);