Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | Aut(G) = C2×#72293994 (order 4096) · Inn(G) = #386 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #123 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
192 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,7,9,5,3)(4,11,12,8,13,16,14,15)(17,18,23,26,22,27,21,25)(19,28,30,24,31,32,29,20);(1,3,6,9,7,10,5,2)(4,11,12,15,13,16,14,8)(17,19)(18,24)(20,27)(21,30)(22,31)(23,29)(25,32)(26,28);(1,4,7,13)(2,8,9,15)(3,11,10,16)(5,14,6,12)(17,20,22,24)(18,19,27,31)(21,28,23,32)(25,30,26,29)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9420, #200770, #200774, … |
| maximal normal subgroup of | C2×#9420, #200770, #200774, … |
| maximal quotient of | #100093, #100094, #100103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2408);