Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,17)(7,18,22,33)(8,24,9,14)(10,28,12,16)(13,23,30,21)(15,26,32,27)(19,31,35,34)(20,29,25,36);(1,3,4,2)(5,7,11,22)(6,13,17,30)(8,25,9,20)(10,23,12,21)(14,31,24,34)(15,29,32,36)(16,33,28,18)(19,27,35,26);(2,3)(5,8,11,9)(6,14,17,24)(7,19,22,35)(10,26,12,27)(13,29,30,36)(15,16,32,28)(18,34,33,31)(20,21,25,23);(5,9,11,8)(6,15,17,32)(7,20,22,25)(10,26,12,27)(13,29,30,36)(14,16,24,28)(18,31,33,34)(19,21,35,23)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25183, #6438852, #6438854, … |
| maximal normal subgroup of | C2×#25183, #6438852, #6438854, … |
| maximal quotient of | #282560, #282561, #282564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18171);