Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | |
764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,5)(3,8,4,7)(10,14)(15,16);(1,3,6,4)(2,7,5,8)(9,10,13,14)(11,15,12,16)(18,22)(23,29)(24,30)(31,32);(1,4,6,3)(2,8,5,7)(9,11,13,12)(10,15,14,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30);(1,2,6,5)(3,7,4,8)(17,19,21,26)(18,23,22,29)(20,27,28,25)(24,32,30,31);(1,5,6,2)(3,8,4,7)(9,12,13,11)(10,16,14,15)(17,20,21,28)(18,24,22,30)(19,25,26,27)(23,31,29,32)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,d^2=gh,e^2=fh,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#35820, #7629540, #7629552, … |
| maximal normal subgroup of | C2×#35820, #7629540, #7629552, … |
| maximal quotient of | #517737, #517739, #517741, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28808);