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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | C22⋊#95 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
602 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)(3,7,12,21)(4,14,16,24)(5,10)(6,9)(8,15,22,17)(11,25)(13,20,28,32)(18,23)(19,29)(26,31)(27,30);(1,3)(2,7,9,21)(4,11)(5,13)(6,12)(8,19,22,30)(10,20,23,32)(14,25,24,31)(15,27)(16,26)(17,29)(18,28);(1,4)(2,8)(3,11)(5,17)(6,16)(7,19)(9,22)(10,24)(12,26)(13,29)(14,23)(15,18)(20,31)(21,30)(25,32)(27,28)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,f^b=fh,e^c=eh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10783, #354322, #355560, … |
| maximal normal subgroup of | C2×#10783, #354322, #355560, … |
| maximal quotient of | #105838, #105839, #105840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3771);