Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | |
688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(10,14)(15,17)(16,21)(22,23)(25,26)(27,29)(28,32)(30,38)(31,34)(33,39)(35,40)(36,37);(10,14)(12,20)(15,17)(18,24)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(1,3,5,8)(2,6,4,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(1,4,5,2)(3,7,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,32)(26,27,34,28)(30,37,39,40)(33,35,38,36);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,39)(26,33,34,38)(27,35,28,36)(29,37,32,40)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=fgh,d^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47525, #7700931, #7700970, … |
| maximal normal subgroup of | C2×#47525, #7700931, #7700970, … |
| maximal quotient of | #576830, #576869, #577069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40513);