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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
710 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)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,26,29,32)(27,33,34,30)(28,31,36,38)(35,39,40,37);(1,2,5,7)(3,4,8,6)(10,14)(11,17)(12,21)(22,24)(27,34)(30,33)(35,40)(37,39);(1,3,5,8)(2,6,7,4)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27)(26,30)(28,35)(29,34)(31,37)(32,33)(36,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,28,29,36)(26,31,32,38)(27,35,34,40)(30,37,33,39)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=fgh,d^2=f,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×#57456, #8100997, #8101197, … |
| maximal normal subgroup of | C2×#57456, #8100997, #8101197, … |
| maximal quotient of | #586550, #587240, #587248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50444);