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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
973 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)(7,10)(8,11)(9,14)(12,17)(13,15)(16,19)(18,20)(21,22,26,24)(23,29,31,27)(25,32,33,28)(30,34,36,35);(6,7)(8,15)(9,17)(18,19)(22,24)(27,29)(28,32)(34,35);(1,2)(3,4)(5,7,10,6)(8,11,15,13)(9,14,17,12)(16,18,20,19)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(21,24,26,22)(23,29,31,27)(25,32,33,28)(30,35,36,34);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^8|a^2=fh,c^2=f,d^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 | #133776, #246337, #246377, … |
| maximal normal subgroup of | #133776, #246337, #246377, … |
| maximal quotient of | #576622, #576675, #576706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39806);