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 | |
692 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)(9,10,14,18)(11,12,16,15)(13,17,21,24)(19,20,23,22)(26,27)(29,37)(31,33)(35,40);(11,16)(12,15)(13,21)(17,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,18)(13,19)(14,16)(17,22)(20,24)(21,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3,5,4)(2,7,6,8)(9,12,14,15)(10,16,18,11)(13,20,21,22)(17,23,24,19)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,18,24)(11,19,16,23)(12,20,15,22)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=g,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×#49370, #7704719, #7704976, … |
| maximal normal subgroup of | C2×#49370, #7704719, #7704976, … |
| maximal quotient of | #560803, #561049, #561923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42358);