Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33469 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = C2×#442 · Out(G) = C24 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#442 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 149 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 — 4 generators on 32 points:
(2,7,9,22)(3,11)(4,14)(5,18)(8,24,25,32)(10,15,19,23)(12,29)(13,16)(17,20)(21,28,26,27);(1,2)(3,8)(4,15)(5,19)(6,9)(7,20)(10,18)(11,28)(12,25)(13,24)(14,22)(16,32)(17,23)(21,30)(26,31)(27,29);(1,3,5,11,6,12,18,29)(2,8,19,28,9,25,10,27)(4,16,14,31,17,13,20,30)(7,21,15,32,22,26,23,24);(3,12)(8,25)(11,29)(13,16)(21,26)(24,32)(27,28)(30,31)Polycyclic presentation:
2^8|a^2=f,c^2=fg,e^2=g,g^2=h,b^a=be,c^a=cg,e^a=eg,g^a=gh,e^b=egh,f^b=fh,g^b=gh,d^c=dh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #497430, #497431, #497868, … |
| maximal normal subgroup of | #497430, #497431, #497868, … |
| maximal quotient of | #483592, #483593, #483597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26457);