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 | |
751 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,26)(21,23)(22,25)(27,31)(28,32)(29,30);(1,2,4,5)(3,8,7,6)(11,16)(13,15)(17,19,23,20)(18,24,21,26)(22,28,30,27)(25,32,29,31);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31);(1,3,4,7)(2,6,5,8)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=g,d^2=fgh,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×#58421, #8104336, #8104422, … |
| maximal normal subgroup of | C2×#58421, #8104336, #8104422, … |
| maximal quotient of | #585629, #586020, #586366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51409);