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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
758 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)(11,15)(13,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28);(2,5)(3,8)(9,10,12,14)(11,13,15,16)(19,27)(21,31)(23,32)(25,28);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32);(1,3,4,8)(2,6,5,7)(9,10)(11,13)(12,14)(15,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^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×#54071, #8099158, #8099326, … |
| maximal normal subgroup of | C2×#54071, #8099158, #8099326, … |
| maximal quotient of | #585264, #585791, #586751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47059);