Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C2×C8×D4 | C22 |
| socle | C22 | #1282 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8×D4 ▷ C22×C4 ▷ C22 ▷ C1 | |
251 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 60 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)(9,10,13,18,16,17,24,22,15,19,14,11,12,21,23,20);(2,5,7,6)(9,11,14,17,16,20,23,19,15,18,13,21,12,22,24,10)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=g,e^2=g,g^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=dfhi,d^c=dh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 205);