Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1790 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
854 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 854 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10)(11,16,20,14)(12,15)(13,18)(17,22)(19,24,21,23);(1,2,4,6)(3,7,8,5)(10,14,18,16)(15,24,17,23);(1,3,4,8)(2,5,6,7)(10,15,18,17)(14,23,16,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ehi,g^b=ghi,d^c=dhi,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234065, #72234084, #72234087, … |
| maximal normal subgroup of | #72234065, #72234084, #72234087, … |
| maximal quotient of | #72234061, #72260619, #72261825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56680);