Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 65536 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
814 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,12,16,14)(17,18,22,26)(19,23,25,21)(20,24,30,32)(27,31,28,29);(2,5)(6,8)(9,11,13,16)(10,14,15,12)(19,25)(21,23)(27,28)(29,31);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,14)(15,16)(17,19,22,25)(18,23,26,21)(20,28,30,27)(24,29,32,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,28,32,27)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fgh,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7005699, #7005711, #7005758, … |
| maximal normal subgroup of | #7005699, #7005711, #7005758, … |
| maximal quotient of | #535450, #535623, #536054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38384);