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 |
| prime radical | #3480 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
918 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)(6,7)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(1,2,4,5)(3,7,8,6);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(1,3,4,8)(2,6,5,7);(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^8|b^2=f,c^2=h,d^2=f,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=df,g^c=gh55 conjugacy classes — view the full character table.
| Fitting subgroup of | #61319⋊C3 |
| maximal subgroup of | #250912, #250944, #250976, … |
| maximal normal subgroup of | #250912, #250944, #250976, … |
| maximal quotient of | Q8×#419, #6379458, #6380337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54307);