Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,13,10,7,14,12)(8,11,17,19,16,15,20,18);(1,2,4,3)(5,7)(6,10)(8,15)(9,12)(11,16)(13,14)(17,18)(19,20);(1,3,4,2)(8,16)(11,15)(17,20)(18,19);(5,8,9,17,10,16,14,20)(6,11,13,19,7,15,12,18)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=h,b^a=be,e^a=eh,e^b=eh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #128570, #128571, #134393, … |
| maximal normal subgroup of | #128570, #128571, #134393, … |
| maximal quotient of | #183874, #183875, #183876, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13500);