Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| 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 | #120 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,11,8,10,13,9,18)(7,14,16,12,17,15,19,20);(1,2,3,4)(5,7)(6,12)(8,15)(9,19)(10,17)(11,16)(13,20)(14,18);(5,8,11,13,10,18,9,6)(7,15,16,20,17,14,19,12)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dh,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10894, #353914, #354311, … |
| maximal normal subgroup of | C2×#10894, #353914, #354311, … |
| maximal quotient of | #109502, #109503, #109504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3882);