Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33657 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#481 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3058 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 283 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,11)(4,14)(5,17)(7,20)(9,24)(10,23)(12,28)(13,16)(15,30)(18,31)(19,29)(21,27)(22,25)(26,32);(1,2)(3,7)(4,15)(5,10)(6,8)(9,18)(11,27)(12,21)(13,22)(14,30)(16,32)(17,23)(19,26)(20,28)(24,31)(25,29);(1,3,5,19,6,12,17,13)(2,7,10,26,8,21,23,22)(4,16,14,28,18,29,31,11)(9,25,24,27,15,32,30,20);(2,8)(7,21)(9,15)(10,23)(20,27)(22,26)(24,30)(25,32)Polycyclic presentation:
2^8|c^2=eg,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^b=fh,e^c=eh,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142696, #142697, #142700, … |
| maximal normal subgroup of | #142696, #142697, #142700, … |
| maximal quotient of | #493921, #494271, #494487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26645);