Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10530 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #124 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10)(3,11,13,16)(4,8,12,7)(5,15,14,9)(17,18,23,26)(19,27,30,31)(20,21,25,22)(24,29,32,28);(1,3,6,13)(2,7,10,8)(4,14,12,5)(9,16,15,11)(17,19,23,30)(18,24,26,32)(20,31,25,27)(21,29,22,28);(1,4,5,3,6,12,14,13)(2,8,9,16,10,7,15,11)(17,20,22,18,23,25,21,26)(19,27,29,24,30,31,28,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=dh,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,e^c=egh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151255, C2×#10535, #238987, … |
| maximal normal subgroup of | #151255, C2×#10535, #238987, … |
| maximal quotient of | #104882, #104889, #104894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3523);