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:
(1,2,5,6)(3,4,7,8)(9,10,13,11)(12,15,16,14)(17,18,22,25)(19,23,26,21)(20,24,30,32)(27,31,28,29);(2,6)(3,7)(9,11,13,10)(12,15,16,14)(19,26)(21,23)(27,28)(29,31);(1,3)(2,4)(5,7)(6,8)(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,28)(21,25)(22,26)(24,29)(27,30)(31,32);(1,4)(2,3)(5,8)(6,7)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(17,21,22,23)(18,19,25,26)(20,29,30,31)(24,27,32,28)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=g,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 | #131526, #131535, #484374, … |
| maximal normal subgroup of | #131526, #131535, #484374, … |
| maximal quotient of | #544562, #545071, #545430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35683);