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) = C26 |
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 | |
593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,12,17,8)(9,15)(10,16)(11,14)(13,19,18,20);(2,4)(5,7,10,13,11,17,9,18)(6,12,15,20,14,8,16,19);(5,8)(6,13)(7,16)(9,20)(10,19)(11,12)(14,18)(15,17)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,f^b=fh,e^c=eh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10784, #353465, C4×#1903, … |
| maximal normal subgroup of | C2×#10784, #353465, C4×#1903, … |
| maximal quotient of | #105845, #105846, #105854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3772);