Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #382 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,16,19,12)(8,21)(9,25)(10,15)(11,14)(13,22)(17,31,33,35)(18,27,20,30)(23,28)(24,36)(26,32,29,34);(2,4)(5,7,10,20,11,19,9,18)(6,12,15,27,14,16,25,30)(8,17,23,34,24,33,22,32)(13,26,21,35,28,29,36,31);(5,8,9,22,11,24,10,23)(6,13,15,21,14,28,25,36)(7,17,18,32,19,33,20,34)(12,26,27,35,16,29,30,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=e,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #146730, #146736, #146740, … |
| maximal normal subgroup of | #146730, #146736, #146740, … |
| maximal quotient of | #108862, #108863, #108864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3856);