Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7422 |
| automorphism | Aut(G) = #72285396 (order 2048) · Inn(G) = #1347 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1347 |
| commutator | C2×C16 | C2×C4 |
| Frattini, cosocle | C2×M32 | C22 |
| prime radical | C23 | SD32 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,12,9,14,8,35)(3,15,17,49,21,50,20,56)(4,22,25,57,28,48,29,58)(6,31,30,47,7,33,24,13)(10,36,37,52,40,53,39,16)(11,41,43,26,45,60,44,27)(18,54,32,63,19,55,34,42)(23,46,59,62,51,64,61,38);(1,3,9,21)(2,10,14,40)(4,23,28,51)(5,20,8,17)(6,32,7,34)(11,42,45,63)(12,37,35,39)(13,46,47,64)(15,48,50,22)(16,26,52,27)(18,24,19,30)(25,61,29,59)(31,62,33,38)(36,41,53,60)(43,54,44,55)(49,58,56,57)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=fgh,d^2=h,e^2=g,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fgh,g^a=gh,c^b=cf,d^b=defgh,e^b=egh,f^b=fgh,g^b=gh,d^c=dh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7425, #150257, #150390, … |
| maximal normal subgroup of | C2×#7425, #150257, #150390, … |
| maximal quotient of | #93787, #93809, #93851, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 413);