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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(10,14)(11,17)(16,21)(19,23)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,3)(2,6)(4,5)(7,8)(12,20)(16,21)(19,23)(22,24)(26,31)(28,32)(34,37)(36,39);(3,4)(6,8)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(1,4,5,3)(2,6,7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,2)(3,8)(4,6)(5,7)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)Polycyclic presentation:
2^8|a^2=h,c^2=fgh,d^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×#56748, #8099538, #8100301, … |
| maximal normal subgroup of | C2×#56748, #8099538, #8100301, … |
| maximal quotient of | #586103, #586254, #586463, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49736);