Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| 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 | |
424 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,21,23)(19,25,27,32)(20,28,30,24)(22,29,31,26);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(17,18)(19,22)(20,28)(21,23)(24,30)(25,29)(26,32)(27,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18445, #6347039, #6347045, … |
| maximal normal subgroup of | C2×#18445, #6347039, #6347045, … |
| maximal quotient of | #169648, #169668, #169712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11433);