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:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,18)(13,15)(14,17)(19,23)(20,24)(21,22)(25,26,31,28)(27,29,32,35)(30,33,39,36)(34,37,38,40);(2,3)(6,7)(9,11,15,12)(10,16,13,18)(14,20,22,19)(17,24,21,23)(25,27,31,32)(26,29,28,35)(30,38,39,34)(33,40,36,37);(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24)(25,28,31,26)(27,29,32,35)(30,36,39,33)(34,37,38,40);(1,3,5,2)(4,7,8,6)(9,13,15,10)(11,18,12,16)(14,21,22,17)(19,24,20,23)(25,29)(26,32)(27,28)(30,37)(31,35)(33,38)(34,36)(39,40);(1,4,5,8)(2,6,3,7)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30)(26,33)(27,34)(28,36)(29,37)(31,39)(32,38)(35,40)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=gh,d^2=fg,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×#57712, #8101914, #8101951, … |
| maximal normal subgroup of | C2×#57712, #8101914, #8101951, … |
| maximal quotient of | #585842, #586296, #587038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50700);