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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,7)(3,5)(10,14)(11,17)(12,20)(22,24)(25,26)(27,31)(28,30)(29,36)(32,37)(33,39)(34,35)(38,40);(1,2,6,7)(3,4,5,8)(26,28)(31,35)(32,36)(38,39);(1,3,6,5)(2,8,7,4)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,34)(26,31,28,35)(29,33,37,40)(32,38,36,39);(1,4,6,8)(2,3,7,5)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,30,26)(27,31,34,35)(29,36,37,32)(33,38,40,39);(1,5,6,3)(2,8,7,4)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29)(26,32)(27,33)(28,36)(30,37)(31,38)(34,40)(35,39)Polycyclic presentation:
2^8|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×#52868, #8092562, #8092708, … |
| maximal normal subgroup of | C2×#52868, #8092562, #8092708, … |
| maximal quotient of | #582278, #582463, #582486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45856);