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:
(1,2,4,6)(3,7,8,5)(9,10,14,12)(11,15,17,19)(13,20,21,16)(18,24,23,22)(25,26,30,34)(27,31,35,40)(28,32,38,37)(29,33,39,36);(1,2,4,6)(3,5,8,7)(10,12)(11,17)(13,21)(22,24)(27,35)(29,39)(31,40)(33,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,39)(34,40)(36,38);(1,3)(2,5)(4,8)(6,7)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(9,13,14,21)(10,16,12,20)(11,18,17,23)(15,22,19,24)(25,29)(26,33)(27,37)(28,31)(30,39)(32,35)(34,36)(38,40)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=f,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×#53983, #8102924, #8103195, … |
| maximal normal subgroup of | C2×#53983, #8102924, #8103195, … |
| maximal quotient of | #585306, #586497, #587202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46971);