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) = C23 |
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 | |
228 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)(9,10,13,18,15,20,14,19)(11,21,23,36,25,38,24,37)(12,26,27,35,28,17,29,33)(16,30,31,22,32,39,34,40);(1,3)(2,5,6,7)(4,8)(9,11,14,24,15,25,13,23)(10,16,19,31,20,32,18,34)(12,22,27,39,28,40,29,30)(17,21,33,37,26,38,35,36)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=h,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 | #93207, #93208, C2×#7306, … |
| maximal normal subgroup of | #93207, #93208, C2×#7306, … |
| maximal quotient of | #93102, #93105, #93108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 294);