Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| 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 | |
3259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(3,9)(4,12)(13,16)(19,25)(22,29)(26,31)(30,32);(1,2,7,6)(3,10,9,5)(4,8,12,14)(11,16,15,13)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,15)(14,16)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31);(1,4)(2,8)(3,11)(5,13)(6,14)(7,12)(9,15)(10,16);(1,5,7,10)(2,9,6,3)(4,13,12,16)(8,15,14,11)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32);(1,6,7,2)(3,10,9,5)(4,14,12,8)(11,16,15,13)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)Polycyclic presentation:
2^8|b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #186159, #186166, #186173, … |
| maximal normal subgroup of | #186159, #186166, #186173, … |
| maximal quotient of | #7626235, #7626236, #7626239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55826);