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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,7)(3,5,4,8)(10,14)(11,17)(16,21)(18,23)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(2,7)(3,4)(10,14)(15,20)(16,21)(22,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(1,3,6,4)(2,8,7,5)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,4,6,3)(2,5,7,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,5)(2,3)(4,7)(6,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^8|a^2=fgh,b^2=gh,c^2=fgh,d^2=f,e^2=fg,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×#57845, #8101461, #8101966, … |
| maximal normal subgroup of | C2×#57845, #8101461, #8101966, … |
| maximal quotient of | #587133, #587144, #587295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50833);