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 | |
744 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)(3,5)(4,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31);(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,14,16,15)(18,21)(19,26)(20,28)(31,32);(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,29)(21,25)(22,26)(24,32)(27,28)(30,31);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,15,16,14)(17,20,22,28)(18,24,21,30)(19,27,26,29)(23,31,25,32);(1,3,4,7)(2,5,6,8)(17,21,22,18)(19,25,26,23)(20,30,28,24)(27,32,29,31)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=gh,d^2=g,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×#44564, #7668843, #7668972, … |
| maximal normal subgroup of | C2×#44564, #7668843, #7668972, … |
| maximal quotient of | #535520, #535721, #536216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37552);