Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
738 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18,23,21)(19,26,24,20)(22,29,31,25)(27,28,30,32);(1,2)(3,7)(4,6)(5,8)(17,19,23,24)(18,20,21,26)(22,30,31,27)(25,32,29,28);(1,2,4,6)(3,5,8,7)(9,10)(11,13)(12,14)(15,16)(17,20,23,26)(18,24,21,19)(22,28,31,32)(25,30,29,27);(1,2)(3,5)(4,6)(7,8)(17,21,23,18)(19,20,24,26)(22,29,31,25)(27,28,30,32);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,22,23,31)(18,25,21,29)(19,27,24,30)(20,28,26,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=f,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130380, #486074, #486568, … |
| maximal normal subgroup of | #130380, #486074, #486568, … |
| maximal quotient of | #576856, #577126, #577338, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40526);