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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,13,15)(11,12,16,14)(18,22)(20,28)(23,25)(26,32);(9,11,13,16)(10,14,15,12)(18,22)(20,28)(23,25)(26,32);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,3,4,7)(2,5,6,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=h,d^2=fg,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×#44495, #7668863, #7669100, … |
| maximal normal subgroup of | C2×#44495, #7668863, #7669100, … |
| maximal quotient of | #535448, #535619, #536048, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37483);