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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,12)(11,17,18,15)(13,16,21,20)(19,23,24,22)(25,26)(27,32)(28,33)(29,36)(30,31)(34,39)(35,38)(37,40);(1,2,5,4)(3,6,7,8)(11,18)(13,21)(15,17)(16,20)(26,31)(27,35)(28,37)(29,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,27,30,35)(26,32,31,38)(28,34,37,36)(29,33,39,40);(1,2,5,4)(3,6,7,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,28,30,37)(26,33,31,40)(27,36,35,34)(29,38,39,32);(1,4,5,2)(3,8,7,6)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29)(26,34)(27,33)(28,38)(30,39)(31,36)(32,37)(35,40)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=f,d^2=fg,e^2=g,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×#47427, #7701358, #7701421, … |
| maximal normal subgroup of | C2×#47427, #7701358, #7701421, … |
| maximal quotient of | #576924, #576977, #577111, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40415);