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 | #2688 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
850 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)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30);(1,3,5,8)(2,6,4,7)(9,10,13,15)(11,12,16,14)(18,20)(19,25)(21,29)(30,32);(2,4)(6,7)(9,11,13,16)(10,14,15,12)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31);(1,4,5,2)(3,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)Polycyclic presentation:
2^8|a^2=f,b^2=fgh,c^2=fh,d^2=fg,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×#57221, #8101112, #8101179, … |
| maximal normal subgroup of | C2×#57221, #8101112, #8101179, … |
| maximal quotient of | #585790, #586028, #586532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50209);