Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12102 |
| automorphism | Aut(G) = C23×#21452 (order 2048) · Inn(G) = #447 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3191 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
174 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 39 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:
(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(18,23)(19,22)(20,27)(24,31)(25,32)(26,29)(28,30);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18)(19,24)(20,28)(21,25)(22,30)(23,27)(26,32)(29,31);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,19,21,22)(18,24,25,30)(20,27,29,26)(23,31,32,28)Polycyclic presentation:
2^8|a^2=gh,c^2=h,d^2=fh,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,d^b=df,e^b=eg,f^b=fh,d^c=dg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12283, #484539, #484542, … |
| maximal normal subgroup of | C2×#12283, #484539, #484542, … |
| maximal quotient of | #126292, #126294, #126303, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5271);