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:
(2,5)(6,8)(9,10,14,17)(11,13,16,22)(12,15,21,24)(18,20,19,23)(25,26,31,29)(27,34,32,28)(30,37,39,33)(35,36,38,40);(11,16)(13,22)(18,19)(20,23)(25,27,31,32)(26,28,29,34)(30,38,39,35)(33,40,37,36);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,13,17,22)(12,19,21,18)(15,23,24,20)(25,28,31,34)(26,32,29,27)(30,36,39,40)(33,38,37,35);(9,12)(10,15)(11,18)(13,20)(14,21)(16,19)(17,24)(22,23)(25,29,31,26)(27,28,32,34)(30,37,39,33)(35,36,38,40);(1,3,4,7)(2,6,5,8)(9,13)(10,16)(11,17)(12,20)(14,22)(15,19)(18,24)(21,23)(25,30,31,39)(26,33,29,37)(27,35,32,38)(28,36,34,40)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,d^2=f,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×#47529, #7700916, #7700931, … |
| maximal normal subgroup of | C2×#47529, #7700916, #7700931, … |
| maximal quotient of | #576987, #577019, #577125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40517);