Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#107 |
| prime radical | C23 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,21,16,23,32,22)(11,24,26,36,27,18,28,40)(12,17,29,37,30,39,25,38)(13,31,33,19,34,20,14,35);(2,5)(6,8)(9,11,14,25,15,26,13,12,16,27,33,29,32,28,34,30)(10,17,20,36,22,38,19,24,23,39,31,40,21,37,35,18);(1,3,4,7)(2,6,5,8)(9,12,13,28,32,25,14,27,16,30,34,26,15,29,33,11)(10,18,19,38,21,40,20,17,23,24,35,37,22,36,31,39)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=d,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #134991, #136399, #136413, … |
| maximal normal subgroup of | #134991, #136399, #136413, … |
| maximal quotient of | #135653, #135673, #135674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5880);