Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(5,6,7,8)(9,10,13,15,14,18,12,17)(11,16,21,22,19,24,20,23);(1,2,4,3)(10,15)(11,19)(12,13)(16,23)(17,18)(22,24);(1,3,4,2)(9,11,12,20,14,19,13,21)(10,16,17,23,18,24,15,22)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,d^2=h,g^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dh,g^b=gh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11290, #358305, #358306, … |
| maximal normal subgroup of | C2×#11290, #358305, #358306, … |
| maximal quotient of | #120700, #120701, #120742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4278);