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 | #2661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
765 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)(3,6)(4,5)(7,8)(9,10,15,16)(11,12,13,14)(17,18,22,25)(19,21,24,30)(20,23,29,32)(26,28,27,31);(2,4)(3,7)(9,11,15,13)(10,12,16,14)(19,24)(21,30)(26,27)(28,31);(1,3,5,7)(2,6,4,8)(9,12,15,14)(10,11,16,13)(17,19,22,24)(18,21,25,30)(20,27,29,26)(23,31,32,28);(1,3)(2,6)(4,8)(5,7)(9,13,15,11)(10,12,16,14)(17,20)(18,23)(19,26)(21,28)(22,29)(24,27)(25,32)(30,31);(1,4,5,2)(3,8,7,6)(9,14,15,12)(10,11,16,13)(17,21)(18,24)(19,25)(20,28)(22,30)(23,27)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=gh,d^2=g,e^2=fg,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×#57132, #8099759, #8100303, … |
| maximal normal subgroup of | C2×#57132, #8099759, #8100303, … |
| maximal quotient of | #585999, #586359, #586895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50120);