Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7457 |
| automorphism | Aut(G) = #483875 (order 512) · Inn(G) = #1368 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1368 |
| commutator | C16 | C2×C8 |
| Frattini, cosocle | #346 | C22 |
| prime radical | C22 | #365 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #346 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,7,8,17,10,19,22,20)(3,11,12,25)(4,14)(5,16)(9,23,21,18,24,31,30,32)(13,28,27,26)(15,29);(1,2,3,7,4,9,11,18,5,8,13,20,14,21,26,23,6,10,12,19,15,24,25,32,16,22,27,17,29,30,28,31)Polycyclic presentation:
2^8|a^2=d,b^2=c,c^2=e,d^2=f,e^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=dgh,f^b=fh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7457, #151024, #151059, … |
| maximal normal subgroup of | C2×#7457, #151024, #151059, … |
| maximal quotient of | #93877, #94030, C2×#7457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 445);