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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10)(11,12)(13,21)(14,15)(16,22)(17,20)(18,24)(19,23)(25,26,30,33)(27,28,31,35)(29,32,37,39)(34,36,38,40);(1,2,5,4)(3,6,7,8)(10,15)(11,17)(13,22)(19,24)(27,31)(28,35)(29,37)(32,39);(1,3)(2,6)(4,8)(5,7)(9,11,14,17)(10,12,15,20)(13,18,22,23)(16,19,21,24)(25,27)(26,28)(29,34)(30,31)(32,36)(33,35)(37,38)(39,40);(1,2,5,4)(3,6,7,8)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,23)(22,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,4,5,2)(3,8,7,6)(9,13)(10,16)(11,18)(12,19)(14,22)(15,21)(17,23)(20,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=f,d^2=g,e^2=g,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×#49401, #7704237, #7704980, … |
| maximal normal subgroup of | C2×#49401, #7704237, #7704980, … |
| maximal quotient of | #560750, #561094, #561837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42389);