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 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3178 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)(7,13)(8,14)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(18,22)(23,25)(24,29)(30,31);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|d^2=h,e^2=h,f^2=g,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 | #6350881, #6350882, #6350905, … |
| maximal normal subgroup of | #6350881, #6350882, #6350905, … |
| maximal quotient of | #7626619, #7626655, #7626661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55938);