Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18930 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3544 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18,22,24)(19,26,23,31)(20,28,29,27)(21,30,32,25);(4,13)(8,12)(9,16)(11,14)(17,19,22,23)(18,20,24,29)(21,28,32,27)(25,31,30,26);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,20)(18,23)(19,24)(21,31)(22,29)(25,27)(26,32)(28,30);(2,7)(6,15)(8,12)(9,16)(18,24)(20,29)(25,30)(26,31)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18933, #6348915, #6348958, … |
| maximal normal subgroup of | C2×#18933, #6348915, #6348958, … |
| maximal quotient of | #170384, #170432, #170492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11921);