Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | #2668 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,4,8,6)(9,10,14,12)(11,15,13,16)(17,18)(19,20)(21,24)(22,23);(1,2,5,7)(3,4,8,6)(10,12)(11,13)(18,23)(20,24);(1,3,5,8)(2,6,7,4)(9,11,14,13)(10,15,12,16)(17,19,22,21)(18,20,23,24);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,15,13,16)(17,20,22,24)(18,19,23,21);(1,4)(2,3)(5,6)(7,8)(9,13,14,11)(10,15,12,16)(17,21,22,19)(18,24,23,20)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#43593, #7667835, #7668084, … |
| maximal normal subgroup of | C2×#43593, #7667835, #7668084, … |
| maximal quotient of | #535433, #535779, #536146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 36581);