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 | |
341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(6,7)(9,10,13,17,14,16,12,21)(11,15,20,23,19,24,18,22);(5,6,8,7)(9,11)(10,15)(12,18)(13,20)(14,19)(16,24)(17,23)(21,22);(2,4)(5,7,8,6)(10,16)(11,18)(12,13)(15,22)(19,20)(23,24)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,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,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11216, #358076, #358077, … |
| maximal normal subgroup of | C2×#11216, #358076, #358077, … |
| maximal quotient of | #120004, #120029, #120030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4204);