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 | |
3681 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)(4,6)(5,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23);(1,2,4,6)(3,5,8,7)(11,17)(15,21)(18,23)(22,24);(1,2)(3,5)(4,6)(7,8)(10,14)(15,21)(16,19)(22,24);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)Polycyclic presentation:
2^8|b^2=g,d^2=g,e^2=h,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 | #347002, #347010, #347014, … |
| maximal normal subgroup of | #347002, #347010, #347014, … |
| maximal quotient of | #7626370, #7626379, #7626383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55868);