Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,13)(8,17)(9,11)(10,16)(12,18)(14,20)(15,19);(5,7,9,14)(6,10,11,19)(8,15,18,16)(12,20,17,13);(2,4)(7,14)(10,19)(13,20)(15,16);(7,15)(8,18)(10,20)(12,17)(13,19)(14,16);(6,11)(10,19)(12,17)(13,20)Polycyclic presentation:
2^8|b^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,c^b=ch,d^b=df,f^b=fh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #497330, #6573736, #6792444, … |
| maximal normal subgroup of | #497330, #6573736, #6792444, … |
| maximal quotient of | #6358587, #6358594, #6358601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54047);