Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,11,9,13,18,12)(7,10,15,14,16,20,17,19);(1,2)(3,4)(5,6)(7,14)(8,12)(9,13)(10,15)(11,18)(16,19)(17,20);(2,4)(5,6)(7,10)(8,11)(9,13)(12,18)(14,15)(16,20)(17,19);(5,7,9,16)(6,10,13,20)(8,17,18,15)(11,19,12,14)Polycyclic presentation:
2^8|a^2=e,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,e^b=eh,f^c=fh,e^d=eh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #6667752, #6667756, #6667764, … |
| maximal normal subgroup of | #6667752, #6667756, #6667764, … |
| maximal quotient of | #333135, #333142, #333150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23465);