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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,26,31,34)(27,32,28,29)(30,38,39,33)(35,37,36,40);(1,2,5,6)(3,4,7,8)(10,14)(15,21)(16,19)(22,23)(25,27,31,28)(26,29,34,32)(30,36,39,35)(33,40,38,37);(2,6)(3,7)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(1,3)(2,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,29,31,32)(26,28,34,27)(30,37,39,40)(33,36,38,35);(1,4)(2,3)(5,8)(6,7)(9,12,13,20)(10,16,14,19)(11,17,18,24)(15,22,21,23)(25,30,31,39)(26,33,34,38)(27,35,28,36)(29,37,32,40)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=f,d^2=g,e^2=f,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×#55081, #8102435, #8102616, … |
| maximal normal subgroup of | C2×#55081, #8102435, #8102616, … |
| maximal quotient of | #585904, #586233, #586708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48069);