Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9758 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15,20,30)(12,22)(13,27)(14,17)(16,25)(18,24)(19,33,23,37)(21,36,34,29)(26,38)(28,39,31,35)(32,40);(1,3,7,2,5,8,4,6)(9,11,14,20)(10,15)(12,23,26,19)(13,21,24,34)(16,31)(17,30)(18,29)(22,37)(25,39,40,35)(27,36)(28,32)(33,38);(2,6)(3,8)(9,12,24,40,14,26,13,25)(10,16,27,38,17,32,18,22)(11,19,34,39,20,23,21,35)(15,28,36,37,30,31,29,33)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eg,d^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,f^b=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94560, #94807, #94810, … |
| maximal normal subgroup of | #94560, #94807, #94810, … |
| maximal quotient of | #102041, #102043, #102044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2748);