Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,13)(14,16)(17,18,23,28)(19,25,20,24)(21,30,31,26)(22,27,32,29);(1,2)(3,7)(4,5)(6,8)(17,19,23,20)(18,24,28,25)(21,29,31,27)(22,30,32,26);(2,5)(6,7)(9,10,12,13)(11,14,15,16)(17,20,23,19)(18,25,28,24)(21,29,31,27)(22,30,32,26);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,29)(20,27)(22,24)(23,31)(25,32)(28,30);(1,3,4,8)(2,6,5,7)(17,22,23,32)(18,27,28,29)(19,26,20,30)(21,24,31,25)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6698372, #6698377, #6698437, … |
| maximal normal subgroup of | #6698372, #6698377, #6698437, … |
| maximal quotient of | #544621, #545154, #545621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35588);