Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,8)(9,10)(11,20)(12,13)(14,22)(15,16)(17,21)(18,24)(19,23);(3,7)(6,8)(10,16)(11,21)(13,18)(19,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,22)(13,23)(14,18)(15,21)(16,20)(19,24);(9,12)(10,18)(11,22)(13,16)(14,17)(15,24)(19,21)(20,23);(1,3,4,7)(2,6,5,8)(9,13,15,18)(10,12,16,24)(11,23,21,14)(17,22,20,19);(1,2)(3,6)(4,5)(7,8)(9,14,15,23)(10,19,16,22)(11,18,21,13)(12,17,24,20)Polycyclic presentation:
2^8|e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #186158, #186165, #186172, … |
| maximal normal subgroup of | #186158, #186165, #186172, … |
| maximal quotient of | #7626233, #7626236, #7626239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55824);