Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,13)(7,15,17,31)(8,12,21,29)(9,22,25,14)(11,23,27,18)(16,33,34,28)(19,36,35,26)(20,32,24,30);(5,7)(6,11)(8,19)(9,23)(10,17)(12,28)(13,27)(14,31)(15,22)(16,20)(18,25)(21,35)(24,34)(26,30)(29,33)(32,36);(2,4);(5,8,10,21)(6,12,13,29)(7,16,17,34)(9,24,25,20)(11,26,27,36)(14,32,22,30)(15,33,31,28)(18,35,23,19);(6,13)(7,17)(12,29)(14,22)(16,34)(18,23)(19,35)(30,32)Polycyclic presentation:
2^8|a^2=h,d^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,d^b=df,e^b=eh,f^b=fh,f^d=fh55 conjugacy classes — view the full character table.
| maximal subgroup of | #6993877, #6993878, #6993888, … |
| maximal normal subgroup of | #6993877, #6993878, #6993888, … |
| maximal quotient of | #6358847, #6358855, #6358863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54064);