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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
689 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)(3,5)(4,7)(6,8)(9,10,14,17)(11,13,16,22)(12,15,21,24)(18,20,19,23)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(2,3)(6,7)(11,16)(13,22)(18,19)(20,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,2,5,3)(4,6,8,7)(9,11,14,16)(10,13,17,22)(12,19,21,18)(15,23,24,20)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,3,5,2)(4,7,8,6)(9,12)(10,15)(11,18)(13,20)(14,21)(16,19)(17,24)(22,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,4,5,8)(2,6,3,7)(9,13)(10,16)(11,17)(12,20)(14,22)(15,19)(18,24)(21,23)(25,29,30,38)(26,32,33,37)(27,35,28,36)(31,39,34,40)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=fgh,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×#56827, #8099980, #8100300, … |
| maximal normal subgroup of | C2×#56827, #8099980, #8100300, … |
| maximal quotient of | #586221, #586413, #586480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49815);