Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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)(9,10,14,15)(11,12,16,19)(13,21,22,17)(18,24,23,20)(26,30)(27,33)(28,35)(38,40);(1,2,5,4)(3,6,7,8)(10,15)(12,19)(17,21)(20,24)(25,26,29,30)(27,31,33,37)(28,36,35,32)(34,40,39,38);(1,3,5,7)(2,6,4,8)(9,11,14,16)(10,12,15,19)(13,18,22,23)(17,20,21,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,27,29,33)(26,31,30,37)(28,34,35,39)(32,38,36,40);(1,4,5,2)(3,8,7,6)(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,28,29,35)(26,32,30,36)(27,34,33,39)(31,38,37,40)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=fg,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46204, #7702751, #7702849, … |
| maximal normal subgroup of | C2×#46204, #7702751, #7702849, … |
| maximal quotient of | #557178, #557231, #557484, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39192);