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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
813 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10,13,15)(11,12,16,14)(19,23)(21,24);(2,6)(3,8)(9,10,13,15)(11,12,16,14)(17,18,20,22)(19,21,23,24);(9,11,13,16)(10,14,15,12)(17,19,20,23)(18,21,22,24);(1,3,5,8)(2,7,6,4)(9,11)(10,12)(13,16)(14,15);(1,4,5,7)(2,8,6,3)(9,12)(10,11)(13,14)(15,16)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,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×#43589, #7667835, #7668072, … |
| maximal normal subgroup of | C2×#43589, #7667835, #7668072, … |
| maximal quotient of | #535432, #535782, #536144, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 36577);