Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| automorphism | Aut(G) = C2×#6346637 (order 1024) · Inn(G) = #3109 · Out(G) = C23 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #2190 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
262 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(3,11)(4,15)(5,19)(7,20)(9,24)(10,23)(12,17)(13,28)(14,29)(16,30)(18,31)(21,32)(22,27)(25,26);(1,2)(3,7)(4,16)(5,10)(6,8)(9,18)(11,27)(12,26)(13,22)(14,21)(15,30)(17,32)(19,23)(20,28)(24,31)(25,29);(1,3)(2,7)(4,17)(5,12)(6,13)(8,22)(9,25)(10,26)(11,15)(14,19)(16,32)(18,29)(20,24)(21,23)(27,30)(28,31);(2,8)(3,12,13,14)(4,18)(7,21,22,26)(10,23)(11,17,28,29)(15,31)(20,32,27,25)Polycyclic presentation:
2^8|d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #486080, #486081, #486082, … |
| maximal normal subgroup of | #486080, #486081, #486082, … |
| maximal quotient of | #494054, #494995, #494996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26679);