Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | |
697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26)(27,34)(28,29)(30,31)(32,37)(33,38)(35,40)(36,39);(1,2,4,6)(3,5,7,8)(10,12)(11,18)(13,21)(22,23)(26,31)(28,33)(32,34)(35,39);(1,3,4,7)(2,5,6,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,27,30,37)(26,32,31,34)(28,35,33,39)(29,36,38,40);(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,28,30,33)(26,29,31,38)(27,35,37,39)(32,36,34,40);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,33)(27,36)(28,31)(30,38)(32,39)(34,35)(37,40)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=g,d^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46852, #7700704, #7700737, … |
| maximal normal subgroup of | C2×#46852, #7700704, #7700737, … |
| maximal quotient of | #576666, #576672, #576758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39840);