Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7304 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #342 |
| commutator | C24 | C42 |
| Frattini, prime radical, cosocle | C22×#41 | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19)(12,23,24,16)(13,17)(14,18)(15,20,27,33)(21,31)(22,32)(25,36,37,29)(26,30)(28,34,38,40)(35,39);(1,3)(2,5,6,7)(4,8)(9,11,13,21,14,22,26,35)(10,15,17,28,18,27,30,38)(12,20,25,34,24,33,37,40)(16,19,29,31,23,32,36,39)Polycyclic presentation:
2^8|a^2=d,b^2=e,e^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=df,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | #93205, #93206, C2×#7305, … |
| maximal normal subgroup of | #93205, #93206, C2×#7305, … |
| maximal quotient of | #93101, #93104, #93107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 293);