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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
482 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)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,18)(19,27)(20,23)(21,26)(22,25)(24,30)(28,31)(29,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(18,23)(20,21)(24,31)(25,26)(27,32)(29,30);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(17,19)(18,24)(20,30)(21,29)(22,28)(23,31)(25,32)(26,27);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)(18,25)(19,28)(23,26)(29,30)Polycyclic presentation:
2^8|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×#18962, #6348941, #6348959, … |
| maximal normal subgroup of | C2×#18962, #6348941, #6348959, … |
| maximal quotient of | #170413, #170473, #170512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11950);