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 | #2678 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,22,26)(19,24,20,23)(21,29,30,25)(27,31,28,32);(2,5)(6,7)(9,11)(10,15)(12,14)(13,16)(17,19,22,20)(18,23,26,24)(21,28,30,27)(25,32,29,31);(11,12)(15,16)(17,20,22,19)(18,24,26,23)(21,28,30,27)(25,32,29,31);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,13,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,32)(28,31)(29,30);(1,3,4,8)(2,6,5,7)(9,13,14,10)(11,15,12,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=h,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×#58158, #8104567, #8104697, … |
| maximal normal subgroup of | C2×#58158, #8104567, #8104697, … |
| maximal quotient of | #585389, #586739, #587037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51146);