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 | |
687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,17)(11,15,18,12)(13,16,21,23)(19,22,24,20)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,3,5,2)(4,7,8,6)(11,18)(12,15)(13,21)(16,23)(26,29)(27,33)(32,37)(34,36);(1,4)(2,6)(3,7)(5,8)(9,11,14,18)(10,15,17,12)(13,19,21,24)(16,22,23,20)(25,27,30,33)(26,31,29,35)(28,36,38,34)(32,40,37,39);(9,12,14,15)(10,11,17,18)(13,20,21,22)(16,19,23,24)(25,28,30,38)(26,32,29,37)(27,34,33,36)(31,39,35,40);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,26)(27,35,33,31)(28,37,38,32)(34,40,36,39)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=h,d^2=h,e^2=f,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×#51429, #7707029, #7707034, … |
| maximal normal subgroup of | C2×#51429, #7707029, #7707034, … |
| maximal quotient of | #560448, #560616, #561455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44417);