Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10631 |
| automorphism | Aut(G) = C23×#7737411 (order 4096) · Inn(G) = #397 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #397 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #124 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
139 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,2,5,7,6,10,4,9)(3,11,13,16,12,15,14,8)(17,18)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(2,7)(3,12)(4,5)(8,15)(9,10)(11,16)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,32,31,29);(1,3,4,13,6,12,5,14)(2,8,9,11,10,16,7,15)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32)Polycyclic presentation:
2^8|a^2=eh,b^2=h,c^2=d,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10636, #242959, #242960, … |
| maximal normal subgroup of | C2×#10636, #242959, #242960, … |
| maximal quotient of | #105223, #105224, #105227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3624);