Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7523 |
| automorphism | Aut(G) = C2×#13569 (order 512) · Inn(G) = #1404 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1404 |
| commutator | #125 | C2×C4 |
| Frattini, cosocle | #490 | C22 |
| prime radical | #127 | D4 |
| LCS | G ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #490 ◁ G | |
| LECS | G ▷ #490 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
67 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,11)(4,15,17,8)(5,10)(6,9,19,27)(7,12,22,30)(13,29)(14,28,20,32)(16,23,31,24)(18,25,21,26);(1,3,5,13)(2,7,10,24)(4,16,18,12)(6,20,19,14)(8,11,26,29)(9,23,27,22)(15,32,25,28)(17,30,21,31)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=g,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=cg,d^b=deh,e^b=egh,f^b=fg,d^c=dfg,e^c=eg,f^c=fh,e^d=egh,f^d=fh19 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7524, #153483, #153484, … |
| maximal normal subgroup of | C2×#7524, #153483, #153484, … |
| maximal quotient of | #94545, #94601, #94663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 512);