Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139962 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #3490 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
684 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 684 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6,8,5)(4,7)(9,10,13,17,14,18,12,15)(11,19,21,16,22,23,20,24);(3,8)(10,15,18,17)(16,19,24,23);(2,5)(3,8)(6,7)(9,11,14,22)(10,16,18,24)(12,20,13,21)(15,19,17,23)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=g,d^2=g,e^2=gi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,h^a=hi,e^b=ehi,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72233230, #72233387, #72233400, … |
| maximal normal subgroup of | #72233230, #72233387, #72233400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47244);