Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8738 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
209 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,10,12,6)(4,9)(5,8)(7,11,14,16)(13,15)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,16)(14,15)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(1,3)(2,7,8,14)(4,11)(5,12)(6,9,10,15)(13,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=d,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94550, #94551, #94552, … |
| maximal normal subgroup of | #94550, #94551, #94552, … |
| maximal quotient of | #99128, #99130, #99133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1728);