Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93021 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1289 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | C2×C8×D4 | C22 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8×D4 ▷ C22×C4 ▷ C22 ▷ C1 | |
257 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 257 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 2 generators on 24 points:
(1,2)(3,6,8,5)(4,7)(10,15,16,19)(11,12)(13,21)(17,24,23,18)(20,22);(2,5)(3,8)(6,7)(9,10,12,17,21,24,11,19,14,16,22,23,13,18,20,15)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fghi,e^2=fgi,f^2=hi,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gi,c^b=ch,d^b=dg,d^c=di56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 221);