Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
659 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)(9,10,13,18,14,17,21,24)(11,16,19,23,12,15,20,22);(2,4)(5,6)(7,8)(9,11,14,12)(10,15,17,16)(13,20,21,19)(18,23,24,22);(6,8)(9,12,14,11)(10,16,17,15)(13,19,21,20)(18,22,24,23)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93116, #93175, #93894, … |
| maximal normal subgroup of | #93116, #93175, #93894, … |
| maximal quotient of | #111209, #111264, #111331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3971);