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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
793 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)(3,4)(5,6)(7,8)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29);(2,6)(3,7)(9,10,13,11)(12,15,16,14)(20,28)(23,30)(26,32)(29,31);(10,11)(14,15)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(1,3,5,7)(2,4,6,8)(9,11,13,10)(12,15,16,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,15)(13,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=fh,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×#58351, #8104005, #8104111, … |
| maximal normal subgroup of | C2×#58351, #8104005, #8104111, … |
| maximal quotient of | #586588, #586804, #587312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51339);