Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C22⋊C8 | C23 |
| socle | C22 | C2×#324 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
508 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,18,13,17,21,24,14,16,20,23,11,15,19,22);(2,5,6,7)(3,8)(10,15,16,17)(12,20)(18,22,23,24)(19,21);(10,16)(15,17)(18,23)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=g,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,d^b=dhi,e^b=efi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29544);