Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12148 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | #1867 |
| prime radical | C2×#120 | Q8 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
416 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2,5,8,6,9,4,13)(3,10,11,7,12,15,14,16)(17,18)(19,26)(20,24)(21,25)(22,28)(23,30)(27,31)(29,32);(1,3,5,11,6,12,4,14)(2,7,8,15,9,16,13,10)(17,19)(18,22)(20,28)(21,29)(23,32)(24,26)(25,31)(27,30);(17,20,21,30)(18,23)(19,27,29,28)(22,26)(24,25)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,d^c=dh,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43306);