Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,16)(11,14,12,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,2)(3,7)(4,5)(6,8)(9,11,13,12)(10,14,16,15)(18,22)(23,25)(24,29)(30,31);(2,5)(6,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,2)(3,6)(4,5)(7,8)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,15,16,14)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41020, #7639361, #7639590, … |
| maximal normal subgroup of | C2×#41020, #7639361, #7639590, … |
| maximal quotient of | #544759, #545249, #545795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34008);