Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,15,9,14,11,16)(7,17,20,32,19,12,26,28)(8,21,24,30,23,31,25,13)(18,34,22,27,35,29,36,33);(1,2,4,3)(5,7,9,19)(6,12,14,17)(8,22,23,36)(10,20,11,26)(13,29,30,34)(15,28,16,32)(18,24,35,25)(21,33,31,27);(5,8,9,23)(6,13,14,30)(7,18,19,35)(10,25,11,24)(12,27,17,33)(15,31,16,21)(20,36,26,22)(28,34,32,29);(1,3,4,2)(6,14)(12,17)(13,30)(15,16)(21,31)(27,33)(28,32)(29,34)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24775, #6443028, #6443030, … |
| maximal normal subgroup of | C2×#24775, #6443028, #6443030, … |
| maximal quotient of | #276960, #276966, #276972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17763);