Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7389 |
| automorphism | Aut(G) = #72284874 (order 1024) · Inn(G) = #1333 · Out(G) = C23 |
| 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 | C2×D16 | 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 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,11,6,13,17,31,8,14,23,12,7,25,5,24)(3,15,16,29,19,9,18,26,21,27,22,30,20,28,32,10);(2,9)(3,16)(4,5)(6,7)(10,24)(11,30)(12,29)(13,15)(14,28)(17,23)(18,20)(19,32)(21,22)(25,27)(26,31)Polycyclic presentation:
2^8|a^2=d,c^2=eh,d^2=f,e^2=g,f^2=h,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×#7390, #150074, #150129, … |
| maximal normal subgroup of | C2×#7390, #150074, #150129, … |
| maximal quotient of | #93757, #93766, #93922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 378);