Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | Aut(G) = C22×#6346637 (order 2048) · Inn(G) = #327 · Out(G) = C25 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #327 |
| commutator | C2×C8 | C2×C8 |
| Frattini, cosocle | C2×#97 | C22 |
| prime radical | C23 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5,6,7)(9,10,11,16,14,18,22,15,12,19,21,23,13,17,20,24);(1,2)(3,7)(4,6)(5,8)(10,15,19,24)(11,20)(13,14)(16,17,23,18)(21,22)Polycyclic presentation:
2^8|a^2=d,b^2=eh,c^2=f,d^2=g,f^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=defh,f^b=fh,g^b=gh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7127, #124098, #124162, … |
| maximal normal subgroup of | C2×#7127, #124098, #124162, … |
| maximal quotient of | #92874, #92878, #92880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 115);