Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(17,18,23,26)(19,24,20,21)(22,30,31,25)(27,29,28,32);(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,19,23,20)(18,21,26,24)(22,28,31,27)(25,32,30,29);(2,5)(6,8)(10,13)(14,15)(17,20,23,19)(18,24,26,21)(22,28,31,27)(25,32,30,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,24)(18,20,26,19)(22,29,31,32)(25,28,30,27);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,22,23,31)(18,25,26,30)(19,27,20,28)(21,29,24,32)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#53451, #8102435, #8102612, … |
| maximal normal subgroup of | C2×#53451, #8102435, #8102612, … |
| maximal quotient of | #585482, #585822, #586048, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46439);