Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7412 |
| automorphism | Aut(G) = C23×#2127 (order 1024) · Inn(G) = #354 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #354 |
| commutator | C2×C42 | C2×C4 |
| Frattini, cosocle | #391 | C22 |
| prime radical | C23 | #99 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
89 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)(3,5)(4,7)(6,8)(9,10,12,17,14,19,25,34)(11,20,22,37,21,35,38,33)(13,16,28,32,29,18,26,36)(15,24,30,27,31,39,40,23);(2,5)(3,8)(6,7)(9,11)(10,15,19,31)(12,24)(13,27)(14,21)(16,20,18,35)(17,33,34,37)(22,28)(23,29)(25,39)(26,38)(30,36,40,32)Polycyclic presentation:
2^8|a^2=d,b^2=g,c^2=fh,d^2=h,e^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=def,e^b=eh,d^c=dgh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7413, #150213, #150214, … |
| maximal normal subgroup of | C2×#7413, #150213, #150214, … |
| maximal quotient of | #93791, #93805, #93832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 401);