Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19328 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
285 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32)Polycyclic presentation:
2^8|a^2=h,b^2=e,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^c=dg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139198, #139201, #489433, … |
| maximal normal subgroup of | #139198, #139201, #489433, … |
| maximal quotient of | #189590, #197846, #202831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12348);