Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,6)(3,4)(7,8)(10,14)(11,12)(15,16);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15);(1,2)(3,6)(4,7)(5,8)(11,12)(14,16);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^8|d^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fgh40 conjugacy classes — view the full character table.
| Fitting subgroup of | #19073⋊C3, #72340786, #72340796 |
| maximal subgroup of | #152626, #153175, #153179, … |
| maximal normal subgroup of | #152626, #153175, #153179, … |
| maximal quotient of | #168115, #168156, #168205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12061);