Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | Aut(G) = C4×#33553 (order 1024) · Inn(G) = #353 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #353 |
| commutator | C2×M16 | C2×C4 |
| Frattini, cosocle | #407 | C22 |
| socle | C2 | #1352 |
| prime radical | C24 | Q16 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
91 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,9,6,11,13,10)(3,12,15,29,17,30,16,25)(4,18,20,26,22,28,21,8)(7,19,23,31,24,32,27,14);(1,3,13,15,6,17,5,16)(2,7,9,23,11,24,10,27)(4,19,20,31,22,32,21,14)(8,25,28,12,26,29,18,30)Polycyclic presentation:
2^8|a^2=d,b^2=eg,c^2=fgh,d^2=h,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=cf,d^b=defgh,e^b=eh,g^b=gh,d^c=dgh,e^c=eh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7433, #150281, #150326, … |
| maximal normal subgroup of | C2×#7433, #150281, #150326, … |
| maximal quotient of | #93245, #93246, #93247, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 421);