Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #340 |
| prime radical | C23 | #107 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,7,5,8,12,15)(3,9,11,16,10,6,13,14)(18,23)(19,27)(20,29)(21,28)(24,25)(26,32)(30,31);(1,3)(2,6,8,9)(4,11)(5,10)(7,14,15,16)(12,13)(17,18,20,24,21,25,19,23,22,26,27,31,28,30,29,32)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=f,f^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7288, #127872, #127873, … |
| maximal normal subgroup of | C2×#7288, #127872, #127873, … |
| maximal quotient of | #93406, #93429, #93457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 276);