Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | |
691 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)(6,8)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26,30,33)(27,31,28,35)(29,37,38,32)(34,40,36,39);(1,2)(3,6)(4,5)(7,8)(10,15)(12,16)(17,22)(20,23)(25,27,30,28)(26,31,33,35)(29,34,38,36)(32,39,37,40);(2,5)(6,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28,30,27)(26,31,33,35)(29,36,38,34)(32,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,39)(36,40)(37,38);(1,3)(2,6)(4,7)(5,8)(9,13,14,21)(10,17,15,22)(11,18,19,24)(12,20,16,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52469, #8092280, #8092466, … |
| maximal normal subgroup of | C2×#52469, #8092280, #8092466, … |
| maximal quotient of | #582273, #582461, #582489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45457);