Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | #2813 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(10,15)(11,13)(17,18)(19,27)(20,25)(21,26)(22,23)(24,30)(28,29)(31,32);(9,10,14,15)(11,12,13,16)(18,23)(20,26)(24,27)(28,32);(2,5)(6,7)(9,11,14,13)(10,16,15,12)(17,19,22,30)(18,24,23,27)(20,28,26,32)(21,29,25,31);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,13)(17,20,22,26)(18,25,23,21)(19,28,30,32)(24,31,27,29);(1,3,4,8)(2,6,5,7)(9,13,14,11)(10,16,15,12)(17,21,22,25)(18,26,23,20)(19,29,30,31)(24,32,27,28)Polycyclic presentation:
2^8|a^2=fg,b^2=h,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52875, #8092562, #8092708, … |
| maximal normal subgroup of | C2×#52875, #8092562, #8092708, … |
| maximal quotient of | #582278, #582466, #582484, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45863);