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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10,13,12)(11,15,16,14)(17,18,22,25)(19,21,26,23)(20,24,30,32)(27,29,28,31);(9,11,13,16)(10,14,12,15)(19,26)(21,23)(27,28)(29,31);(1,2,4,5)(3,6,7,8)(17,19)(18,23)(20,28)(21,25)(22,26)(24,29)(27,30)(31,32);(9,12,13,10)(11,14,16,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,19,25,26)(20,29,30,31)(24,27,32,28)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=gh,d^2=f,e^2=g,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×#44639, #7668883, #7669105, … |
| maximal normal subgroup of | C2×#44639, #7668883, #7669105, … |
| maximal quotient of | #535473, #535754, #536156, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37627);