Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | Aut(G) = #72315303 (order 2048) · Inn(G) = #391 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #391 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1985 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
154 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,7,3,6,11,24,15,8,12,25,16,5,10,23,14)(4,17,20,29,19,9,26,13,21,27,31,18,22,28,32,30);(1,3,7,10,6,15,24,2,8,16,25,11,5,14,23,12)(4,18,31,28,19,30,32,17,21,29,20,9,22,13,26,27);(1,4,6,19,8,21,5,22)(2,9,10,17,12,28,11,27)(3,13,14,29,16,30,15,18)(7,20,24,26,25,31,23,32)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=e,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cg,e^b=eh,f^b=fh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132079, #132080, C2×#13269, … |
| maximal normal subgroup of | #132079, #132080, C2×#13269, … |
| maximal quotient of | #141865, #141866, #141870, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6257);