Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) = #72278809 (order 2048) · Inn(G) = #99 · Out(G) = C22×M16 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C22×C4 | C2×C8 |
| Frattini, cosocle | C2×C4⋊C8 | C22 |
| socle | C22 | #324 |
| prime radical | C23 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,6,8,5)(4,7)(9,10,12,17,13,18,23,31,14,16,22,28,24,29,35,38)(11,15,20,25,21,27,34,36,19,26,32,30,33,37,40,39);(2,5,7,6)(9,11,14,19)(10,15)(12,20,22,32)(13,21,24,33)(16,26)(17,30)(18,27)(23,34,35,40)(25,28)(29,37)(31,39)(36,38)Polycyclic presentation:
2^8|a^2=d,b^2=eg,c^2=gh,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=ch,d^b=degh,d^c=dg46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7066, #122328, #122329, … |
| maximal normal subgroup of | C2×#7066, #122328, #122329, … |
| maximal quotient of | #92872, #92991, #93738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54);