Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(6,7)(9,10,12,17,14,18,13,16)(11,15,21,19,22,24,20,23);(5,6,8,7)(9,11)(10,15)(12,21)(13,20)(14,22)(16,23)(17,19)(18,24);(2,4)(5,7,8,6)(9,10)(11,19)(12,17)(13,16)(14,18)(15,20)(21,24)(22,23)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,e^b=eh,f^b=fh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11221, #358096, #358097, … |
| maximal normal subgroup of | C2×#11221, #358096, #358097, … |
| maximal quotient of | #120006, #120007, #120034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4209);