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×#538 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,13)(8,12)(9,10)(11,14)(15,20)(16,18)(17,19)(21,22,27,25)(23,30,24,28)(26,33,34,29)(31,35,32,36);(6,9)(7,14)(8,16)(19,20)(21,23,27,24)(22,28,25,30)(26,31,34,32)(29,35,33,36);(1,2,3,4)(5,7)(6,11)(8,17)(9,13)(10,14)(12,20)(15,16)(18,19)(21,24,27,23)(22,28,25,30)(26,32,34,31)(29,35,33,36);(5,8,10,16)(6,12,9,18)(7,15,14,17)(11,19,13,20)(21,25,27,22)(23,28,24,30)(26,33,34,29)(31,35,32,36);(5,9,10,6)(7,13,14,11)(8,18,16,12)(15,20,17,19)(21,26,27,34)(22,29,25,33)(23,31,24,32)(28,35,30,36)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^2=fg,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 | #133778, #246361, #246401, … |
| maximal normal subgroup of | #133778, #246361, #246401, … |
| maximal quotient of | #576616, #576672, #576705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40206);