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 | #3535 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,4)(3,11,15,27)(5,18,20,30)(7,9,22,16)(8,23,12,13)(10,25,17,21)(14,26,29,31)(19,24,32,28);(1,3,7,14)(2,8,9,24)(4,12,16,28)(5,19,21,13)(6,15,22,29)(10,26,18,11)(17,31,30,27)(20,32,25,23);(1,4)(2,6)(3,12)(5,10)(7,16)(8,15)(9,22)(11,13)(14,28)(17,20)(18,21)(19,26)(23,27)(24,29)(25,30)(31,32);(2,9)(4,16)(8,24)(10,18)(11,26)(12,28)(17,30)(27,31)Polycyclic presentation:
2^8|a^2=h,b^2=g,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×#18263, #6346569, #6346584, … |
| maximal normal subgroup of | C2×#18263, #6346569, #6346584, … |
| maximal quotient of | #167004, #167024, #167068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11251);