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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,11)(12,15,20,17)(13,18,21,16)(19,23,24,22)(25,26,30,33)(27,34,35,31)(28,32,38,29)(36,40,39,37);(1,2,5,4)(3,6,7,8)(10,11)(13,21)(15,17)(19,24)(27,35)(28,38)(29,32)(31,34);(1,3)(2,6)(4,8)(5,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27)(26,31)(28,39)(29,37)(30,35)(32,40)(33,34)(36,38);(1,2,5,4)(3,6,7,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28)(26,32)(27,36)(29,33)(30,38)(31,37)(34,40)(35,39);(1,4,5,2)(3,8,7,6)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,32)(26,28,33,38)(27,37,35,40)(31,39,34,36)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=f,d^2=gh,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×#51302, #7707242, #7707285, … |
| maximal normal subgroup of | C2×#51302, #7707242, #7707285, … |
| maximal quotient of | #560810, #561068, #561930, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44290);