Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,13,10,14,8,18)(7,15,16,11,17,20,19,12);(1,2)(3,4)(5,7)(6,11)(8,19)(9,16)(10,17)(12,14)(13,20)(15,18);(2,4)(5,7)(6,12)(8,16)(9,19)(10,17)(11,14)(13,20)(15,18);(5,6,10,14)(7,12,17,11)(8,18,9,13)(15,19,20,16)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^c=fh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24469, #6437930, #6437932, … |
| maximal normal subgroup of | C2×#24469, #6437930, #6437932, … |
| maximal quotient of | #268216, #268218, #268219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17457);