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 | |
727 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,11)(12,15,16,14)(17,18,22,25)(19,26,27,23)(20,21,24,31)(28,29,30,32);(2,4)(6,7)(9,11,13,10)(12,15,16,14)(19,27)(20,24)(21,31)(23,26);(1,2,5,4)(3,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,19,22,27)(18,23,25,26)(20,30,24,28)(21,29,31,32);(1,3,5,8)(2,6,4,7)(17,20,22,24)(18,21,25,31)(19,28,27,30)(23,32,26,29);(1,4,5,2)(3,7,8,6)(17,21)(18,24)(19,29)(20,25)(22,31)(23,28)(26,30)(27,32)Polycyclic presentation:
2^8|a^2=fg,b^2=gh,c^2=h,d^2=h,e^2=f,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×#51483, #7707344, #7707345, … |
| maximal normal subgroup of | C2×#51483, #7707344, #7707345, … |
| maximal quotient of | #560435, #560634, #561455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44471);