Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #393 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,15,17,19)(8,11)(9,13)(10,14)(12,16,20,18);(2,4)(6,11,14,13)(12,15,20,19);(5,7)(6,12,14,20)(8,18)(9,16)(10,17)(11,19,13,15)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^c=dh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11387, #358798, #358836, … |
| maximal normal subgroup of | C2×#11387, #358798, #358836, … |
| maximal quotient of | #121996, #121997, #121998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4375);