Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | Aut(G) = #72285671 (order 2048) · Inn(G) = #1365 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1365 |
| commutator | C2×C16 | C2×C4 |
| Frattini, cosocle | C4×C16 | C22 |
| prime radical | C22 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
51 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,11,6,24,3,15,5,23,16,27,17,32,18,10,8,14,22,9,26,13,20,31,7,12,21,30,25,28,19,29);(2,9,14,11)(3,16)(4,18)(5,7)(6,25)(10,13,29,24)(12,30,23,27)(15,28,31,32)(17,26)(19,22)(20,21)Polycyclic presentation:
2^8|a^2=d,b^2=eg,c^2=fgh,d^2=efg,e^2=h,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=def,e^b=eh,f^b=fg,g^b=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7453, #150962, #150977, … |
| maximal normal subgroup of | C2×#7453, #150962, #150977, … |
| maximal quotient of | #93940, #94364, C2×#7453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 441);