Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×#120 |
| prime radical | #3475 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18,23,27)(19,28,20,31)(21,25,22,26)(24,29,32,30);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,23,20)(18,24,27,32)(21,29,22,30)(25,28,26,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,19)(18,24,27,32)(21,30,22,29)(25,28,26,31);(17,21,23,22)(18,25,27,26)(19,29,20,30)(24,28,32,31)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,e^b=eg,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #210267, #210275, #210283, … |
| maximal normal subgroup of | #210267, #210275, #210283, … |
| maximal quotient of | #165724, #165763, #165805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11511);