Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7389 |
| automorphism | Aut(G) = #483875 (order 512) · Inn(G) = #1333 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1333 |
| commutator | C16 | C2×C8 |
| Frattini, cosocle | #346 | C22 |
| prime radical | #508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #346 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
59 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,7,8,22,10,24,11,25)(3,12,15,28)(4,17)(5,19)(9,13,18,29,21,30,27,23)(14,26)(16,32,31,20);(1,2)(3,13)(4,18)(5,8)(6,10)(7,20)(9,26)(11,19)(12,22)(14,27)(15,30)(16,23)(17,21)(24,32)(25,28)(29,31)Polycyclic presentation:
2^8|a^2=d,c^2=eh,d^2=f,e^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ceg,d^b=dg,e^b=egh,f^b=fh,g^b=gh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7389, #150071, #150146, … |
| maximal normal subgroup of | C2×#7389, #150071, #150146, … |
| maximal quotient of | #93757, #93758, #93766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 377);