Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#453 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,11,14)(7,16,17,19)(8,18,9,12)(10,15,20,13);(1,3,4,2)(5,7)(6,8)(9,14)(10,18)(11,17)(12,20)(13,19)(15,16);(2,3)(5,8,11,9)(6,12,14,18)(7,13,17,15)(10,16,20,19);(5,9,11,8)(6,7,14,17)(10,16,20,19)(12,13,18,15)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=h,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25187, #6438852, #6438853, … |
| maximal normal subgroup of | C2×#25187, #6438852, #6438853, … |
| maximal quotient of | #282576, #282577, #282580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18175);