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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
693 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)(3,7)(4,5)(6,8)(10,14)(11,17)(16,21)(19,23)(25,26,30,33)(27,31,36,40)(28,29,32,39)(34,38,37,35);(2,5)(6,7)(12,20)(16,21)(19,23)(22,24)(25,27,30,36)(26,31,33,40)(28,34,32,37)(29,38,39,35);(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27)(26,31)(28,37)(29,35)(30,36)(32,34)(33,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,30,32)(26,29,33,39)(27,34,36,37)(31,38,40,35);(1,3,4,8)(2,6,5,7)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29)(26,32)(27,35)(28,33)(30,39)(31,34)(36,38)(37,40)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,d^2=gh,e^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×#58555, #8104210, #8104426, … |
| maximal normal subgroup of | C2×#58555, #8104210, #8104426, … |
| maximal quotient of | #585702, #586166, #586860, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51543);