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 | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(10,14)(15,18)(16,21)(22,24)(25,26)(27,30)(28,31)(29,34)(32,37)(33,36)(35,39)(38,40);(1,3,5,4)(2,6,8,7)(10,14)(11,17)(12,20)(22,24)(26,27)(29,37)(31,33)(35,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4,5,3)(2,7,8,6)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^8|a^2=gh,b^2=gh,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×#51312, #7706990, #7707039, … |
| maximal normal subgroup of | C2×#51312, #7706990, #7707039, … |
| maximal quotient of | #560826, #560955, #561926, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44300);