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 | |
683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(9,10,14,17)(11,18,19,15)(12,16,22,13)(20,24,23,21)(25,26,31,29)(27,34,28,32)(30,37,38,33)(35,39,36,40);(11,19)(12,22)(13,16)(15,18)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,23)(13,21)(14,19)(16,24)(17,18)(20,22)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,20)(13,17)(14,22)(15,21)(18,24)(19,23)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(1,4,5,3)(2,8,6,7)(9,13,14,16)(10,12,17,22)(11,21,19,24)(15,23,18,20)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^8|a^2=fh,b^2=f,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×#51003, #7706610, #7706731, … |
| maximal normal subgroup of | C2×#51003, #7706610, #7706731, … |
| maximal quotient of | #560776, #561130, #561896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43991);