Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18866 |
| 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 | #3518 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
432 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)(8,14)(10,16)(13,15)(17,18,23,28)(19,29,25,32)(20,30,24,31)(21,26,22,27);(1,3,5,10)(2,6,7,15)(4,13,14,11)(8,16,12,9)(17,19,23,25)(18,24,28,20)(21,31,22,30)(26,29,27,32);(17,20,23,24)(18,25,28,19)(21,29,22,32)(26,31,27,30);(2,7)(3,10)(8,12)(11,13)(17,21,23,22)(18,26,28,27)(19,30,25,31)(20,29,24,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=h,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18921, #6348887, #6348888, … |
| maximal normal subgroup of | C2×#18921, #6348887, #6348888, … |
| maximal quotient of | #167520, #167535, #167550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11909);