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×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
796 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)(3,5)(4,7)(6,8)(17,18,22,25)(19,23,20,27)(21,24,29,32)(26,30,28,31);(2,3)(4,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,20)(18,23,25,27)(21,28,29,26)(24,31,32,30);(2,3)(6,7)(10,14)(11,15)(17,20,22,19)(18,23,25,27)(21,28,29,26)(24,30,32,31);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,12,14,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,31)(29,32);(1,4,5,8)(2,6,3,7)(9,12)(10,15)(11,14)(13,16)(17,21,22,29)(18,24,25,32)(19,26,20,28)(23,30,27,31)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=fg,d^2=fh,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×#58157, #8104684, #8104697, … |
| maximal normal subgroup of | C2×#58157, #8104684, #8104697, … |
| maximal quotient of | #585514, #586819, #586834, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51145);