Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7515 |
| automorphism | Aut(G) = C2×#512862 (order 1024) · Inn(G) = #1402 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1402 |
| commutator, prime radical | #127 | C2×C4 |
| Frattini, cosocle | #492 | C22 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,6,11,18,27)(3,12,13,25,15,22,29,32)(5,8,16,24,19,10,17,26)(7,20,21,30,23,28,31,14);(1,3,6,15)(2,7,11,23)(4,13,18,29)(5,14,19,30)(8,12,10,22)(9,25,27,32)(16,20,17,28)(21,24,31,26)Polycyclic presentation:
2^8|a^2=d,b^2=h,d^2=h,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,d^b=degh,e^b=egh,f^b=fgh,d^c=dfg,e^c=egh,f^c=fh,e^d=egh,f^d=fh22 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7518, #153441, #153442, … |
| maximal normal subgroup of | C2×#7518, #153441, #153442, … |
| maximal quotient of | #94539, #94597, #94673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 506);