Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) = C23×#72236189 (order 8192) · Inn(G) = #386 · Out(G) = C27 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,9,14,10,13,8,18)(7,11,17,19,15,20,16,12);(5,7)(6,11,13,20)(8,17)(9,16)(10,15)(12,18,19,14);(1,2,3,4)(5,7)(6,12,13,19)(8,16)(9,17)(10,15)(11,18,20,14)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10954, #354124, C4×#1904, … |
| maximal normal subgroup of | C2×#10954, #354124, C4×#1904, … |
| maximal quotient of | #110129, #110130, #110138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3942);