Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(6,10)(8,16)(11,19)(12,17)(13,20)(14,18);(5,6)(7,13)(8,17)(9,12)(10,16)(11,14)(15,19)(18,20);(2,4);(5,7,9,15)(6,11,12,20)(8,18,16,14)(10,19,17,13);(6,12)(10,17)(11,20)(13,19)Polycyclic presentation:
2^8|d^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,d^b=df,e^b=eh,f^b=fh,f^d=fh55 conjugacy classes — view the full character table.
| maximal subgroup of | #6533070, #6533073, #6533082, … |
| maximal normal subgroup of | #6533070, #6533073, #6533082, … |
| maximal quotient of | #6358725, #6358732, #6358735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54056);