Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | #2673 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28);(3,7)(6,8)(11,16)(13,15)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32);(1,3)(2,6)(4,7)(5,8)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#54016, #8100882, #8101175, … |
| maximal normal subgroup of | C2×#54016, #8100882, #8101175, … |
| maximal quotient of | #585608, #586937, #587266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47004);