Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1437 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7,12,22)(4,14)(5,17)(6,9)(8,16)(10,19)(11,25,27,28)(13,24,18,21)(15,29)(20,31,32,26)(23,30);(1,3)(2,7,9,22)(4,11)(5,18)(6,12)(8,20,23,32)(10,24,17,21)(13,19)(14,28,29,25)(15,27)(16,31)(26,30);(1,4,5,16,6,15,19,30)(2,8,10,14,9,23,17,29)(3,11,18,31,12,27,13,26)(7,20,24,28,22,32,21,25)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10951, #354322, #355560, … |
| maximal normal subgroup of | C2×#10951, #354322, #355560, … |
| maximal quotient of | #110134, #110135, #110141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3939);