Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139956 |
| 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×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 | |
548 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 548 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,7)(3,11,13,8,14,15,12,16)(17,18)(19,22)(20,27,29,23)(21,24)(25,32)(26,30,28,31);(2,7,10,9)(8,11,16,15)(17,19)(18,22)(20,28)(21,25)(23,31)(24,32)(26,29)(27,30);(1,3,6,14)(2,8,10,16)(4,12,5,13)(7,11,9,15)(18,23)(19,25)(22,30)(24,27)(26,28)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47240);