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 | |
789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10,15,16)(11,12,13,14)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32);(1,3)(2,6)(4,5)(7,8)(9,11,15,13)(10,12,16,14)(18,23)(20,24)(26,29)(28,31);(3,4)(6,8)(9,12,15,14)(10,11,16,13)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28);(1,4,5,3)(2,6,7,8)(9,13,15,11)(10,12,16,14)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(1,2)(3,8)(4,6)(5,7)(9,14,15,12)(10,11,16,13)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fgh,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×#57128, #8100133, #8100303, … |
| maximal normal subgroup of | C2×#57128, #8100133, #8100303, … |
| maximal quotient of | #586261, #586415, #587149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50116);