Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) order 65536 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
698 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)(4,7)(6,8)(10,14)(11,17)(16,21)(18,23)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(2,3)(4,8)(10,14)(15,20)(16,21)(22,24)(25,27,31,29)(26,32,28,33)(30,37,38,35)(34,40,36,39);(2,3)(6,7)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,3,5,2)(4,7,8,6)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(1,4,5,8)(2,6,3,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^8|a^2=fgh,b^2=fgh,c^2=h,d^2=fh,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 | #132765, #487519, #488228, … |
| maximal normal subgroup of | #132765, #487519, #488228, … |
| maximal quotient of | #585529, #586677, #587653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51808);