Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,14)(15,16)(17,18,22,24)(19,25,27,20)(21,29,30,23)(26,28,32,31);(2,4)(3,7)(17,19,22,27)(18,20,24,25)(21,26,30,32)(23,28,29,31);(1,2,5,4)(3,6,7,8)(9,10)(11,15)(12,16)(13,14);(1,3,5,7)(2,6,4,8)(9,11,13,12)(10,15,14,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,16,14,15)(17,21,22,30)(18,23,24,29)(19,26,27,32)(20,28,25,31)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130360, #133712, #486036, … |
| maximal normal subgroup of | #130360, #133712, #486036, … |
| maximal quotient of | #557311, #557519, #557770, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39334);