Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | #1867 |
| prime radical | C22×D4 | #41 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
342 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 342 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,7,11,5,13)(3,12,9,8,15,16,14,4)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31);(1,3,6,9,7,15,5,14)(2,8,10,16,11,4,13,12)(18,22)(23,30)(24,31)(25,32);(1,4)(2,9)(3,13)(5,16)(6,12)(7,8)(10,15)(11,14)(17,19)(18,23,24,25)(20,28)(21,27)(22,30,31,32)(26,29)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=hi,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=hi,d^b=dg,e^b=ehi,d^c=dh,e^c=egi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43489);