Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | Aut(G) = C2×#72263997 (order 2048) · Inn(G) = #327 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #327 |
| commutator | C22×C8 | C2×C4 |
| Frattini, cosocle | C2×C22⋊C8 | C22 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,9)(4,12,13,7)(5,8)(6,10,14,16)(11,15)(17,18,20,25,22,27,21,26)(19,28,30,24,31,23,29,32);(1,3,5,11)(2,6,8,14)(4,10,13,16)(7,9,12,15)(18,23)(19,29)(20,21)(24,25)(26,32)(27,28)(30,31)Polycyclic presentation:
2^8|a^2=d,b^2=gh,c^2=fh,d^2=h,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,c^b=cf,d^b=defh,e^b=eh,f^b=fh,d^c=dgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7087, #122619, #122621, … |
| maximal normal subgroup of | C2×#7087, #122619, #122621, … |
| maximal quotient of | #92848, #92881, #93032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 75);