Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | Aut(G) = #72257321 (order 1024) · Inn(G) = #387 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #387 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1917 |
| prime radical | C23 | #131 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
123 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 123 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,13,7,11,25,30,8,12,26,14,5,21,23,17)(3,4,10,20,15,9,27,28,16,19,29,24,22,18,32,31);(1,3,6,10,7,15,25,27,8,16,26,29,5,22,23,32)(2,9,13,28,11,19,30,24,12,18,14,31,21,4,17,20);(1,4,5,18,8,19,7,9)(2,10,11,27,12,29,21,32)(3,14,15,17,16,13,22,30)(6,24,23,28,26,20,25,31)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=d,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cg,d^b=dh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13108, #494781, #494957, … |
| maximal normal subgroup of | C2×#13108, #494781, #494957, … |
| maximal quotient of | #140086, #140092, #140095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6096);