Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | Aut(G) = #72279901 (order 2048) · Inn(G) = #331 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #331 |
| commutator | C42 | C2×C8 |
| Frattini, cosocle | C43 | C22 |
| socle | C2 | #1270 |
| prime radical | #532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,10,11,16,12,15,13,6)(4,7)(5,9)(8,14);(1,3)(2,6)(4,13)(5,12)(7,15)(8,10)(9,16)(11,14)Polycyclic presentation:
2^8|a^2=d,c^2=f,d^2=g,e^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=defh,e^b=eh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #94542, #94543, #94598, … |
| maximal normal subgroup of | #94542, #94543, #94598, … |
| maximal quotient of | #92817, #92818, #92825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 94);