Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12307 |
| automorphism | Aut(G) = C2×#6346637 (order 1024) · Inn(G) = #1616 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1616 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | #469 | C22 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
140 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 35 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:
(1,2,7,11,6,12,5,9)(3,13,15,30,17,31,16,27)(4,18,21,28,20,26,22,10)(8,23,24,14,25,19,29,32);(1,3)(2,8)(4,19)(5,15)(6,17)(7,16)(9,24)(10,27)(11,29)(12,25)(13,26)(14,21)(18,31)(20,23)(22,32)(28,30);(2,9,12,11)(4,20)(8,24,25,29)(10,26,28,18)(13,30,31,27)(14,32)(19,23)(21,22)Polycyclic presentation:
2^8|a^2=fg,c^2=gh,d^2=f,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gh,d^b=dfh,e^b=egh,f^b=fh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12309, #484988, #484989, … |
| maximal normal subgroup of | C2×#12309, #484988, #484989, … |
| maximal quotient of | #126351, #126354, #126390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5297);