Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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)(18,23)(19,26)(21,29)(28,31);(1,2)(3,7)(4,6)(5,8)(17,18,22,23)(19,20,26,24)(21,30,29,25)(27,31,32,28);(1,2,4,6)(3,5,8,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28);(1,2)(3,5)(4,6)(7,8)(17,20,22,24)(18,19,23,26)(21,28,29,31)(25,27,30,32);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)Polycyclic presentation:
2^8|a^2=gh,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 | C2×#47537, #7700916, #7700944, … |
| maximal normal subgroup of | C2×#47537, #7700916, #7700944, … |
| maximal quotient of | #576844, #576860, #577105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40525);