Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C23 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
343 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 127 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)(19,25)(20,22)(21,29)(23,28)(24,26)(27,31)(30,32);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(18,22)(23,30)(24,29)(25,31);(17,19,26,32,20,27,21,28)(18,23,29,31,22,30,24,25);(3,10)(6,15)(9,16)(11,14)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=fh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #143479, #143481, #143491, … |
| maximal normal subgroup of | #143479, #143481, #143491, … |
| maximal quotient of | #191438, #206115, #206118, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12624);