Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139980 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #3492 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
554 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 554 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:
(1,2,3,12,18,14,4,20)(5,11,24,29,22,31,23,27)(6,13,19,8,7,15,16,10)(9,26,28,17,30,25,32,21);(2,8,13,12)(3,16)(4,19)(5,22)(9,28,29,31)(10,15,20,14)(11,30,32,27)(17,23)(21,24)(25,26);(1,3,7,16)(2,9,15,27)(4,6,19,18)(5,21,26,23)(8,11,20,28)(10,31,12,32)(13,29,14,30)(17,25,24,22)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=gi,e^2=gi,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,h^a=hi,c^b=ci,d^b=di,e^b=ehi,f^b=fi,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47237);