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 | |
756 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)(3,7,8,6)(10,13)(11,15)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30);(2,5)(3,8)(9,10,12,13)(11,14,15,16)(21,29)(24,31)(26,32)(28,30);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,14,13,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=fg,d^2=fg,e^2=g,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×#56224, #8103110, #8103191, … |
| maximal normal subgroup of | C2×#56224, #8103110, #8103191, … |
| maximal quotient of | #585324, #585742, #586809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49212);