Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
917 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,22)(18,24)(21,23);(1,2)(3,6)(4,5)(7,8)(11,18)(14,23)(17,24)(19,21);(9,11,13,18)(10,14,15,23)(12,21,22,19)(16,24,20,17);(2,5)(6,8)(10,15)(14,23)(16,20)(17,24);(1,3,4,7)(2,6,5,8)(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^8|c^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=eh,g^a=gh,c^b=ch,d^b=df,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #6358812, #6358816, #6358838, … |
| maximal normal subgroup of | #6358812, #6358816, #6358838, … |
| maximal quotient of | #6379555, #6379558, #6380401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54455);