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 | #3547 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
433 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18,23,26)(19,29,25,32)(20,30,24,31)(21,28,22,27);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19,23,25)(18,24,26,20)(21,31,22,30)(27,32,28,29);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,20,23,24)(18,25,26,19)(21,29,22,32)(27,30,28,31);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(18,26)(20,24)(27,28)(29,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,d^2=gh,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×#18977, #6348954, #6348959, … |
| maximal normal subgroup of | C2×#18977, #6348954, #6348959, … |
| maximal quotient of | #170428, #170464, #170521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11965);