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 | |
700 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(28,36)(31,38)(34,40)(37,39);(2,4)(3,7)(11,18)(14,17)(19,24)(21,23)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,3,5,7)(2,6,4,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,4,5,2)(3,8,7,6)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=f,d^2=f,e^2=g,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×#52884, #8092588, #8092709, … |
| maximal normal subgroup of | C2×#52884, #8092588, #8092709, … |
| maximal quotient of | #582258, #582483, #582485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45872);