Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) = #72279294 (order 2048) · Inn(G) = #99 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C22×C4 | C2×C8 |
| Frattini, cosocle | C2×#97 | C22 |
| socle | C22 | #323 |
| prime radical | C23 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5,6,7)(9,10,11,15,12,16,18,22,13,14,17,20,19,21,23,24);(1,2,4,6)(3,7,8,5)(10,14)(16,21)Polycyclic presentation:
2^8|a^2=d,b^2=f,c^2=f,d^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=def,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #93198, #93199, #93200, … |
| maximal normal subgroup of | #93198, #93199, #93200, … |
| maximal quotient of | #92886, #92887, #92891, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 121);