Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| 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 | |
1348 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)(5,6,11,16)(7,17,20,31)(8,13,23,30)(9,15,10,14)(12,24,27,19)(18,33,35,28)(21,36,34,26)(22,32,25,29);(5,7,11,20)(6,12,16,27)(8,21,23,34)(9,19,10,24)(13,28,30,33)(14,17,15,31)(18,25,35,22)(26,32,36,29);(2,4);(5,8)(6,13)(7,18)(9,22)(10,25)(11,23)(12,26)(14,29)(15,32)(16,30)(17,33)(19,34)(20,35)(21,24)(27,36)(28,31);(5,9,11,10)(6,14,16,15)(7,19,20,24)(8,22,23,25)(12,17,27,31)(13,29,30,32)(18,34,35,21)(26,33,36,28)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,d^b=df,f^b=fh,f^d=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #130737, #130739, #220530, … |
| maximal normal subgroup of | #130737, #130739, #220530, … |
| maximal quotient of | #6358434, #6358443, #6358455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54045);