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 | |
707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,16)(11,14,12,15)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32);(2,5)(3,7)(9,11,13,12)(10,14,16,15)(18,23)(19,26)(25,28)(27,32);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32);(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,15,16,14)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=fh,d^2=f,e^2=g,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 | #6549148, #6549150, #6549162, … |
| maximal normal subgroup of | #6549148, #6549150, #6549162, … |
| maximal quotient of | #521694, #522311, #522830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31231);