Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| 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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,16)(14,17)(15,18)(19,20);(1,2,3,4)(6,10)(11,13)(12,17)(18,19);(6,11)(9,16)(10,13)(12,18)(15,20)(17,19);(5,7)(6,12)(8,14)(9,15)(10,17)(11,18)(13,19)(16,20);(7,14)(12,17)(15,20)(18,19)Polycyclic presentation:
2^8|b^2=fh,g^2=h,b^a=bf,c^a=cg,g^a=gh,g^c=gh,e^d=eh61 conjugacy classes — view the full character table.
| Fitting subgroup of | #60430⋊C3 |
| maximal subgroup of | #6573985, #6573986, #6573990, … |
| maximal normal subgroup of | #6573985, #6573986, #6573990, … |
| maximal quotient of | #6356009, #6356010, #6356011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53418);