Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20286 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×D4 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,4)(3,9,10,16)(5,13,15,8)(7,14,12,11)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32);(1,3,6,10)(2,7,4,12)(5,14,15,11)(8,16,13,9)(17,19)(18,22)(20,27)(21,26)(23,31)(24,30)(25,29)(28,32);(3,10)(7,12)(9,16)(11,14)(18,23)(19,26)(20,28)(22,29)(24,25)(30,31);(1,4,6,2)(3,9,10,16)(5,8,15,13)(7,14,12,11)(18,24)(22,30)(23,25)(29,31)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,c^b=cgh,d^b=de,e^b=eg,f^c=fh,e^d=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20297, #6361239, #6361267, … |
| maximal normal subgroup of | C2×#20297, #6361239, #6361267, … |
| maximal quotient of | #220917, #220931, #220933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13285);