Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139970 |
| 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 | |
610 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 610 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:
(1,2,5,11,22,29,4,20)(3,14,17,9,24,26,15,31)(6,12,21,8,7,13,23,10)(16,28,18,25,32,27,19,30);(2,8,12,11)(4,21)(5,23)(9,14,27,25)(10,13,20,29)(15,32)(16,17)(26,28,30,31);(1,3,7,19)(2,9,13,28)(4,15,21,32)(5,17,23,16)(6,18,22,24)(8,14,20,30)(10,26,11,25)(12,27,29,31)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=gi,e^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,d^b=di,e^b=ehi,f^b=fi,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72238486, #72238488, #72238493, … |
| maximal normal subgroup of | #72238486, #72238488, #72238493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47229);