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 | #2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
729 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)(3,5)(4,6)(7,8)(9,10,14,15)(11,12,19,16)(13,21,22,17)(18,23,24,20)(26,30)(27,33)(28,36)(38,40);(4,8)(6,7)(10,15)(12,16)(13,22)(18,24)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(2,3)(6,7)(9,11,14,19)(10,16,15,12)(13,18,22,24)(17,23,21,20)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,3,5,2)(4,7,8,6)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,4,5,8)(2,6,3,7)(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=fh,d^2=g,e^2=g,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×#57850, #8101648, #8101957, … |
| maximal normal subgroup of | C2×#57850, #8101648, #8101957, … |
| maximal quotient of | #586574, #586769, #587131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50838);