Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| 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 | |
3548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(11,17)(15,18)(19,23)(22,24);(2,5)(6,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh73 conjugacy classes — view the full character table.
| maximal subgroup of | #6533247, #6533253, #6533259, … |
| maximal normal subgroup of | #6533247, #6533253, #6533259, … |
| maximal quotient of | #7626621, #7626633, #7626636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55942);