Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C2×#120 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #538 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(6,12)(13,16)(17,18,21,23)(19,25,27,32)(20,28,30,24)(22,29,31,26);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,18)(19,22)(20,28)(21,23)(24,30)(25,29)(26,32)(27,31);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18283, #6346582, #6346588, … |
| maximal normal subgroup of | C2×#18283, #6346582, #6346588, … |
| maximal quotient of | #167036, #167048, #167099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11271);