Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| 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 | #1667 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(4,12)(7,13)(8,9)(10,16)(14,15)(17,18,22,25)(19,23,21,29)(20,27,28,31)(24,30,32,26);(1,2,5,9)(3,7,11,15)(4,8,12,6)(10,14,16,13)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(1,3,5,11)(2,7,9,15)(4,10,12,16)(6,13,8,14)(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27);(2,8)(4,12)(6,9)(7,14)(10,16)(13,15)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=de,e^b=eg,f^c=fh,e^d=eg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19841, #6358196, #6358237, … |
| maximal normal subgroup of | C2×#19841, #6358196, #6358237, … |
| maximal quotient of | #220751, #220857, #222495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12829);