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 | #2828 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
727 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(28,36)(31,38)(34,40)(37,39);(2,5)(6,8)(10,12)(11,18)(16,20)(19,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,2,4,5)(3,6,7,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=g,d^2=g,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×#52835, #8092698, #8092705, … |
| maximal normal subgroup of | C2×#52835, #8092698, #8092705, … |
| maximal quotient of | #582305, #582508, #582535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45823);