Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| 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 | #3536 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,17)(11,19,21,16)(12,13,18,15)(20,24,23,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,23)(13,24)(14,21)(15,22)(17,19)(18,20);(9,12,15,10,14,18,13,17)(11,20,22,19,21,23,24,16);(3,8)(5,7)(11,21)(16,19)(20,23)(22,24);(2,6)(5,7)(9,13,14,15)(10,12,17,18)(11,22,21,24)(16,20,19,23)Polycyclic presentation:
2^8|a^2=h,c^2=g,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #265352, #265353, #265356, … |
| maximal normal subgroup of | #265352, #265353, #265356, … |
| maximal quotient of | #6386754, #6389146, #6389147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55269);