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 |
| prime radical | #3460 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,9)(7,11)(10,15)(14,16)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31);(1,2,6,5)(3,7,9,11)(4,8,12,13)(10,14,15,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31);(1,5,6,2)(3,11,9,7)(4,8,12,13)(10,14,15,16);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=g,d^2=g,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 | #347073, #347097, #347121, … |
| maximal normal subgroup of | #347073, #347097, #347121, … |
| maximal quotient of | #7626606, #7626635, #7626690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55920);