Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C8×C22⋊C4 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,7)(4,6)(5,8)(9,10,13,16,14,17,21,23,12,11,15,18,20,19,22,24);(1,3)(2,5,6,7)(4,8)(10,11)(16,18)(17,19)(23,24);(9,11,13,18,14,19,21,24,12,10,15,16,20,17,22,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,e^2=h,h^2=i,b^a=bd,f^a=fg,c^b=ci,d^b=dg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35640);