Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19304 |
| 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 | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
299 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16);(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31);(3,10)(6,15)(9,16)(11,14)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32)Polycyclic presentation:
2^8|a^2=gh,b^2=e,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #268287, #268294, #268317, … |
| maximal normal subgroup of | #268287, #268294, #268317, … |
| maximal quotient of | #189124, #197812, #197813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12303);