Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | D4×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1371 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)(5,6,10,7,9,11,17,13)(8,12,16,14,15,18,20,19);(2,4)(6,11)(7,13)(12,18)(14,19);(5,7,10,11,9,13,17,6)(8,14,16,18,15,19,20,12);(5,8)(6,12)(7,14)(9,15)(10,16)(11,18)(13,19)(17,20);(8,15)(12,18)(14,19)(16,20)Polycyclic presentation:
2^8|a^2=g,c^2=g,g^2=h,b^a=bf,c^a=ch,c^b=ch,e^d=eh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #60147⋊C3 |
| maximal subgroup of | #129515, #210276, #210277, … |
| maximal normal subgroup of | #129515, #210276, #210277, … |
| maximal quotient of | #6345913, #6345914, #6345915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53135);