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 | |
795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(10,13)(11,15)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29);(1,3,5,8)(2,6,7,4)(10,13)(14,16)(20,28)(23,30)(26,32)(29,31);(2,7)(4,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(1,4,5,6)(2,3,7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,3)(2,6)(4,7)(5,8)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=fgh,d^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×#56639, #8099664, #8100290, … |
| maximal normal subgroup of | C2×#56639, #8099664, #8100290, … |
| maximal quotient of | #585427, #586101, #586414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49627);