Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10,13,11)(12,15,16,14)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32);(2,6)(3,8)(9,11,13,10)(12,15,16,14)(18,21)(23,25)(24,28)(31,32);(1,3)(2,4)(5,8)(6,7)(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31);(1,4)(2,3)(5,7)(6,8)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29);(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^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 | #131527, #131534, #484376, … |
| maximal normal subgroup of | #131527, #131534, #484376, … |
| maximal quotient of | #544561, #545071, #545430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35685);