Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2826 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2826 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
246 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(1,2,7,9,6,12,5,11)(3,13,14,21,17,23,16,8)(4,10,18,24,20,27,19,26)(15,28,29,25,30,32,31,22);(1,3,5,16,6,17,7,14)(2,8,9,13,12,21,11,23)(4,15,18,29,20,30,19,31)(10,25,26,28,27,22,24,32);(2,9)(3,14)(4,18)(5,7)(8,21)(11,12)(15,30)(16,17)(19,20)(22,28)(24,26)(25,32);(1,4)(2,10)(3,15)(5,19)(6,20)(7,18)(8,22)(9,24)(11,26)(12,27)(13,28)(14,29)(16,31)(17,30)(21,25)(23,32)Polycyclic presentation:
2^8|a^2=gh,b^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,c^b=cg,d^b=def,f^b=fh,d^c=de,e^c=eh,f^c=fh,g^c=gh,e^d=eh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#32871, #6631455, #6631456, … |
| maximal normal subgroup of | C2×#32871, #6631455, #6631456, … |
| maximal quotient of | #391209, #391250, #391251, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25859);