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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,8)(9,10,13,12)(11,15,16,14)(17,18,22,23)(19,20,25,24)(21,26,30,31)(27,28,32,29);(1,2)(3,4)(5,8)(6,7)(9,11,13,16)(10,14,12,15)(18,23)(19,25)(26,31)(27,32);(1,3,6,8)(2,4,7,5)(9,10,13,12)(11,14,16,15)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28);(1,4,6,5)(2,3,7,8)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32);(1,5,6,4)(2,8,7,3)(9,12,13,10)(11,15,16,14)(17,21,22,30)(18,26,23,31)(19,27,25,32)(20,28,24,29)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#36056, #7629527, #7629545, … |
| maximal normal subgroup of | C2×#36056, #7629527, #7629545, … |
| maximal quotient of | #517956, #517957, #517963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29044);