Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12948 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1864 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1864 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
192 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8,11,10)(3,13,14,31)(4,16)(5,20)(6,19)(9,26)(12,21,23,17)(15,22,24,18)(25,32)(27,29);(1,2,6,23,7,11,19,12)(3,9,24,28,14,26,15,30)(4,17,16,8,5,21,20,10)(13,25,22,32,31,29,18,27);(1,3)(2,9)(4,18)(5,22)(6,24)(7,14)(8,25)(10,29)(11,26)(12,30)(13,16)(15,19)(17,27)(20,31)(21,32)(23,28)Polycyclic presentation:
2^8|a^2=f,b^2=fg,d^2=g,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,d^b=dg,e^b=eg,f^b=fh,g^b=gh,d^c=dg,e^c=egh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12948, #494319, #495737, … |
| maximal normal subgroup of | C2×#12948, #494319, #495737, … |
| maximal quotient of | #135904, #135928, #136222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5936);