Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,26,30,28)(27,31,34,35)(29,32,38,36)(33,39,37,40);(10,15)(12,16)(18,21)(20,23)(25,27,30,34)(26,31,28,35)(29,37,38,33)(32,40,36,39);(1,2)(3,6)(4,7)(5,8)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,27)(26,31)(28,35)(29,33)(30,34)(32,39)(36,40)(37,38);(1,3,5,4)(2,6,8,7)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28,30,26)(27,31,34,35)(29,36,38,32)(33,39,37,40);(1,4,5,3)(2,7,8,6)(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,29)(26,32)(27,33)(28,36)(30,38)(31,39)(34,37)(35,40)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=f,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#51296, #7707259, #7707292, … |
| maximal normal subgroup of | C2×#51296, #7707259, #7707292, … |
| maximal quotient of | #560885, #560996, #561994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44284);