Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | Aut(G) = C23×#72236189 (order 8192) · Inn(G) = #386 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3)(5,6,9,14,10,12,8,13)(7,11,16,18,17,19,15,20);(1,2,4,3)(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,20)(14,18);(1,3,4,2)(6,12)(7,15)(8,9)(11,18)(16,17)(19,20)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cf,f^b=fh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11214, #358072, #358178, … |
| maximal normal subgroup of | C2×#11214, #358072, #358178, … |
| maximal quotient of | #120002, #120003, #120029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4202);