Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12093 |
| 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×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | #1867 |
| prime radical | C22×D4 | C2×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 | |
329 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 329 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) = 128 |
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)(19,23)(20,22)(21,24)(25,32)(26,29)(27,31)(28,30);(1,3,5,11,6,12,4,14)(2,7,8,15,9,16,13,10)(18,22)(19,25)(23,31)(24,30)(26,27)(29,32);(17,19,21,25)(18,23,24,32)(20,27,28,26)(22,29,30,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,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^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43295);