Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10,13,12)(11,14,16,15)(18,20)(23,25)(24,28)(30,31);(9,11)(10,14)(12,15)(13,16)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30);(1,2)(3,7)(4,8)(5,6)(9,10,13,12)(11,15,16,14)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(1,3,5,4)(2,7,6,8)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30);(1,4,5,3)(2,8,6,7)(9,12,13,10)(11,14,16,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=h,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #98049, #166909, #166910, … |
| maximal normal subgroup of | #98049, #166909, #166910, … |
| maximal quotient of | #521455, #522218, #522614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32563);