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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
948 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)(10,15)(11,18)(12,14)(16,23)(19,20)(22,24);(1,3)(2,5)(4,8)(6,7)(9,10,13,15)(11,16,20,22)(12,21,14,17)(18,24,19,23);(9,11,14,18,13,20,12,19)(10,16,17,24,15,22,21,23);(3,8)(5,7)(10,15)(16,22)(17,21)(23,24);(2,6)(5,7)(9,12,13,14)(10,17,15,21)(11,19,20,18)(16,24,22,23)Polycyclic presentation:
2^8|b^2=h,c^2=g,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130288, #265340, #265346, … |
| maximal normal subgroup of | #130288, #265340, #265346, … |
| maximal quotient of | #6386748, #6389135, #6389165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55257);