Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10208 |
| automorphism | Aut(G) = #72299933 (order 4096) · Inn(G) = #392 · Out(G) = #392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3214 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
237 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,10)(4,14)(5,8)(6,9)(7,11)(12,15)(13,16)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32);(1,3,6,12)(2,7,9,16)(4,13,5,11)(8,15,14,10)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,31)(28,32);(3,11)(4,5)(7,15)(8,14)(10,16)(12,13)(17,19)(18,23)(20,29)(21,26)(22,28)(24,32)(25,31)(27,30)Polycyclic presentation:
2^8|b^2=g,d^2=g,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fh,d^c=dh,e^c=egh,f^c=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10209, #232817, #232819, … |
| maximal normal subgroup of | C2×#10209, #232817, #232819, … |
| maximal quotient of | #102686, #102687, #102690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3197);