Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2529 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)(5,6,9,13)(7,11,14,18)(8,10,16,12)(15,17,20,19);(6,10)(8,16)(11,17)(12,13)(15,20)(18,19);(2,4);(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20);(7,14)(11,18)(15,20)(17,19)Polycyclic presentation:
2^8|a^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,e^d=eh61 conjugacy classes — view the full character table.
| Fitting subgroup of | #60373⋊C3 |
| maximal subgroup of | #249896, #249900, #249904, … |
| maximal normal subgroup of | #249896, #249900, #249904, … |
| maximal quotient of | #6350062, #6350067, #6350072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53361);