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×#117 | C23 |
| socle | C22 | #1867 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
341 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,7)(3,8)(4,5)(9,10,14,19,15,20,12,17)(11,21,22,38,24,36,25,40)(13,27,29,35,30,18,26,37)(16,31,32,28,33,39,34,23);(1,2)(3,4)(5,7)(6,8)(9,11,14,22,15,24,12,25)(10,16,19,32,20,33,17,34)(13,28,29,39,30,23,26,31)(18,36,37,40,27,21,35,38);(1,3,6,7)(2,4,8,5)(9,12,15,14)(10,17,20,19)(11,22,24,25)(13,29,30,26)(16,32,33,34)(18,35,27,37)(21,38,36,40)(23,39,28,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,d^c=dh,e^c=egi,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43359);