Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | Aut(G) = #72279635 (order 2048) · Inn(G) = #333 · Out(G) = #120 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #333 |
| commutator | C42 | C2×C8 |
| Frattini, cosocle | C43 | C22 |
| socle | C2 | #1272 |
| prime radical | C23 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
115 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)(3,10,12,6,11,15,13,16)(4,7)(5,9)(8,14);(1,3,5,11)(2,6)(4,13,14,12)(7,10)(8,15)(9,16)Polycyclic presentation:
2^8|a^2=d,b^2=f,c^2=fh,d^2=g,e^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=def,e^b=eh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #94548, #94549, C2×#7134, … |
| maximal normal subgroup of | #94548, #94549, C2×#7134, … |
| maximal quotient of | #92888, #92889, #92890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 122);