Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139962 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | |
626 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 626 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,12,4,8)(3,14,16,9,17,25,15,26)(7,13,23,11,20,29,21,24)(18,31,22,27,32,28,19,30);(2,8,12,10)(9,14,26,25)(11,13,24,29)(27,31,30,28);(1,3,7,18)(2,9,13,27)(4,19,21,15)(5,22,23,16)(6,17,20,32)(8,14,24,31)(10,25,11,28)(12,26,29,30)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,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 | #72233403, #72233487, #72233492, … |
| maximal normal subgroup of | #72233403, #72233487, #72233492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47221);