Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| 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:
(2,7,10,9)(3,12,13,31)(4,15)(5,19)(8,23)(11,28,20,16)(14,17,21,32)(18,30)(22,24)(25,27);(1,2,5,20,6,10,19,11)(3,8,21,26,13,23,14,29)(4,16,15,7,18,28,30,9)(12,27,17,25,31,22,32,24);(1,3)(2,8)(4,17)(5,21)(6,13)(7,22)(9,27)(10,23)(11,29)(12,30)(14,19)(15,31)(16,25)(18,32)(20,26)(24,28)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=egh,f^b=fh,g^b=gh,d^c=dgh,e^c=egh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12952, #494329, #495737, … |
| maximal normal subgroup of | C2×#12952, #494329, #495737, … |
| maximal quotient of | #135906, #135907, #135930, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5940);