Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) order 16384 |
| 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 | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,15,18,12)(8,10)(9,14)(11,13)(16,19,20,17);(5,7,11,17,9,18,10,19)(6,12,8,16,14,15,13,20);(1,2,3,4)(5,8,11,14,9,13,10,6)(7,16,17,15,18,20,19,12)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=efh,e^2=h,g^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×#10953, #354122, C4×#1903, … |
| maximal normal subgroup of | C2×#10953, #354122, C4×#1903, … |
| maximal quotient of | #110130, #110131, #110133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3941);