Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10038 |
| automorphism | Aut(G) = C2×#72278822 (order 4096) · Inn(G) = #392 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
233 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 233 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18,22,26)(19,27,30,32)(20,24,21,25)(23,29,31,28);(6,15)(9,16)(17,19,22,30)(18,23,26,31)(20,29,21,28)(24,27,25,32);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(18,24)(19,28)(20,21)(25,26)(27,32)(29,30)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,e^c=eh,f^c=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10044, #235539, #235541, … |
| maximal normal subgroup of | C2×#10044, #235539, #235541, … |
| maximal quotient of | #102472, #102476, #102482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3032);