Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| 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 | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(4,5)(8,15)(9,12)(10,13)(14,16)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32);(1,2,6,9)(3,8,11,16)(4,7,5,12)(10,14,13,15)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,16)(12,15)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(1,4,6,5)(2,7,9,12)(3,10,11,13)(8,14,16,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^8|b^2=g,d^2=h,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19012, #6351398, #6351400, … |
| maximal normal subgroup of | C2×#19012, #6351398, #6351400, … |
| maximal quotient of | #165905, #165907, #165943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12000);