Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12102 |
| automorphism | Aut(G) = C2×#72308187 (order 4096) · Inn(G) = #447 · Out(G) = C22×#41 |
| 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 | C23 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 41 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)(17,18,21,25)(19,24,22,30)(20,28,29,31)(23,26,32,27);(1,4,5,11)(2,7,8,14)(3,9,10,16)(6,12,13,15)(17,19,21,22)(18,24,25,30)(20,27,29,26)(23,31,32,28)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=g,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 | #123065, #123066, #123068, … |
| maximal normal subgroup of | #123065, #123066, #123068, … |
| maximal quotient of | #126326, #126346, #126360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5280);