Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18810 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
429 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 429 conjugacy classes of subgroups — too many to draw. Only its 113 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)(7,11)(12,16)(13,15)(17,18,23,26)(19,29,25,32)(20,30,24,31)(21,28,22,27);(1,3,6,13)(2,7,9,16)(4,11,5,12)(8,15,14,10)(17,19,23,25)(18,24,26,20)(21,31,22,30)(27,32,28,29);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,20,23,24)(18,25,26,19)(21,29,22,32)(27,30,28,31);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(18,26)(19,25)(27,28)(30,31)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,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^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #244609, #244610, #244611, … |
| maximal normal subgroup of | #244609, #244610, #244611, … |
| maximal quotient of | #168948, #168996, #169060, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11837);