Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,13)(7,8,11,16)(9,12,18,20)(14,15,17,19)(21,22,26,31)(23,28,34,27)(24,29,30,25)(32,33,35,36);(5,7,10,11)(6,8,13,16)(9,17,18,14)(12,19,20,15)(21,23,26,34)(22,27,31,28)(24,35,30,32)(25,36,29,33);(1,2,3,4)(5,7)(6,8)(9,14)(10,11)(12,15)(13,16)(17,18)(19,20)(21,23)(22,28)(24,35)(25,33)(26,34)(27,31)(29,36)(30,32);(5,8)(6,11)(7,13)(9,15)(10,16)(12,17)(14,20)(18,19)(21,24,26,30)(22,29,31,25)(23,32,34,35)(27,36,28,33);(5,9)(6,12)(7,14)(8,15)(10,18)(11,17)(13,20)(16,19)(21,25,26,29)(22,30,31,24)(23,33,34,36)(27,35,28,32)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133773, #246363, #246403, … |
| maximal normal subgroup of | #133773, #246363, #246403, … |
| maximal quotient of | #576617, #576675, #576716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40199);