Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19671 |
| automorphism | Aut(G) order 32768 |
| 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 | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
356 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2,6,8)(3,10,12,16)(4,9,5,14)(7,13,15,11)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,3,5,11,6,12,4,13)(2,7,9,10,8,15,14,16)(19,26)(22,31)(25,32)(27,30);(2,8)(7,15)(9,14)(10,16)(17,19,20,27,21,26,29,30)(18,22,23,25,24,31,28,32);(1,4,6,5)(2,9,8,14)(3,11,12,13)(7,16,15,10)Polycyclic presentation:
2^8|a^2=gh,b^2=e,c^2=fg,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=ch,d^b=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | #289204, #289214, #289240, … |
| maximal normal subgroup of | #289204, #289214, #289240, … |
| maximal quotient of | #191447, #206181, #206183, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12674);