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 | |
679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,26,30,28)(27,31,34,35)(29,32,38,36)(33,39,37,40);(10,14)(15,21)(16,19)(22,23)(25,27,30,34)(26,31,28,35)(29,37,38,33)(32,40,36,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,39)(36,40)(37,38);(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,28,30,26)(27,31,34,35)(29,36,38,32)(33,39,37,40);(1,3,4,7)(2,5,6,8)(9,12,13,20)(10,16,14,19)(11,17,18,24)(15,22,21,23)(25,29)(26,32)(27,33)(28,36)(30,38)(31,39)(34,37)(35,40)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=f,d^2=h,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×#51260, #7707259, #7707269, … |
| maximal normal subgroup of | C2×#51260, #7707259, #7707269, … |
| maximal quotient of | #560824, #561147, #561911, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44248);