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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,20)(21,25)(22,23)(24,27)(26,28)(29,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,12,16)(18,23)(20,24)(25,30)(28,31);(2,5)(6,8)(9,12,14,10)(11,15,13,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32);(1,2)(3,6)(4,5)(7,8)(9,13,14,11)(10,15,12,16)(17,20,22,24)(18,19,23,27)(21,28,29,31)(25,26,30,32);(1,3)(2,6)(4,7)(5,8)(17,21,22,29)(18,25,23,30)(19,26,27,32)(20,28,24,31)Polycyclic presentation:
2^8|b^2=fg,c^2=fg,d^2=g,e^2=f,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×#52593, #8092416, #8092471, … |
| maximal normal subgroup of | C2×#52593, #8092416, #8092471, … |
| maximal quotient of | #582326, #582512, #582541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45581);