Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #386 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(5,6)(7,15)(8,11)(9,13)(10,14)(12,16)(17,19)(18,20);(6,11,14,13)(12,19,20,15);(1,3,4,2)(5,7)(6,12)(8,16)(9,18)(10,17)(11,19)(13,15)(14,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10895, #353916, C4×#1906, … |
| maximal normal subgroup of | C2×#10895, #353916, C4×#1906, … |
| maximal quotient of | #109510, #109511, #109514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3883);