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 | |
3251 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)(3,12)(4,14)(5,9)(6,13)(7,8)(10,15)(11,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(2,8)(3,5)(10,14)(11,13)(18,22)(23,25)(24,29)(30,31);(1,3,7,5)(2,9,8,12)(4,11,15,13)(6,10,16,14)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,4,7,15)(2,10,8,14)(3,13,5,11)(6,12,16,9)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,5,7,3)(2,9,8,12)(4,11,15,13)(6,14,16,10)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31);(1,6)(2,11)(3,10)(4,12)(5,14)(7,16)(8,13)(9,15)Polycyclic presentation:
2^8|c^2=h,d^2=h,e^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 | #346491, #346501, #346513, … |
| maximal normal subgroup of | #346491, #346501, #346513, … |
| maximal quotient of | #7626362, #7626364, #7626400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55860);