Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| 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 | |
3575 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23);(2,3)(6,7)(11,17)(15,21)(18,23)(22,24);(1,3,5,2)(4,7,8,6)(10,14)(15,21)(16,19)(22,24);(1,4,5,8)(2,6,3,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh70 conjugacy classes — view the full character table.
| maximal subgroup of | #346485, #346495, #346506, … |
| maximal normal subgroup of | #346485, #346495, #346506, … |
| maximal quotient of | #7626350, #7626356, #7626388, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55854);