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 | |
3197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(18,22)(20,28)(23,25)(26,32);(3,9)(7,13)(10,15)(14,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(2,6)(7,13)(8,11)(14,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)Polycyclic presentation:
2^8|b^2=g,c^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 | #347161, #347171, #347183, … |
| maximal normal subgroup of | #347161, #347171, #347183, … |
| maximal quotient of | #7626368, #7626381, #7626408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55865);