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 | C2×D4 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
282 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)(3,9,13,16)(5,14,15,8)(7,11,10,12)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(1,3,6,13)(2,7,4,10)(5,11,15,12)(8,16,14,9);(1,4,6,2)(3,10,13,7)(5,14,15,8)(9,12,16,11)(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(3,11)(5,15)(7,16)(8,14)(9,10)(12,13)Polycyclic presentation:
2^8|a^2=gh,b^2=g,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×#19816, #6358162, #6358238, … |
| maximal normal subgroup of | C2×#19816, #6358162, #6358238, … |
| maximal quotient of | #220745, #220854, #222483, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12804);