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 | #2673 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
840 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)(3,5)(4,6)(7,8)(17,18,22,26)(19,24,20,23)(21,29,30,25)(27,31,28,32);(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19,22,20)(18,23,26,24)(21,28,30,27)(25,32,29,31);(2,3)(6,7)(10,13)(14,15)(17,20,22,19)(18,24,26,23)(21,28,30,27)(25,32,29,31);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,32)(28,31)(29,30);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^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×#53617, #8103255, #8103313, … |
| maximal normal subgroup of | C2×#53617, #8103255, #8103313, … |
| maximal quotient of | #585796, #586090, #586838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46605);