Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7339 |
| automorphism | Aut(G) = #72286380 (order 4096) · Inn(G) = #1316 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1316 |
| commutator | C22×C4 | C2×C8 |
| Frattini, cosocle | #407 | C22 |
| prime radical | C24 | M16 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,10,6,9,20,25,7,11,19,28,4,17,18,27)(3,12,14,29,15,23,31,26,16,24,30,32,13,8,21,22);(1,3,7,16)(2,8,11,23)(4,15,6,13)(5,14,19,30)(9,24,17,12)(10,26,28,22)(18,31,20,21)(25,29,27,32)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=h,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ch,d^b=deh,f^b=fh,d^c=dg,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94525, #94620, #94621, … |
| maximal normal subgroup of | #94525, #94620, #94621, … |
| maximal quotient of | #93197, #93201, #93674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 331);