Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11915 |
| automorphism | Aut(G) = #72306485 (order 4096) · Inn(G) = #1533 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1533 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #97 | C23 |
| prime radical | C23 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,10,5,11,18,29,6,8,17,25,20,9,26,27)(3,12,13,30,14,7,21,23,15,24,31,28,19,22,16,32);(1,3,6,15)(2,7,8,22)(4,16,17,21)(5,19,20,14)(9,24,11,12)(10,28,25,30)(13,18,31,26)(23,27,32,29);(2,8)(4,17)(7,22)(9,11)(12,24)(13,31)(16,21)(18,26)Polycyclic presentation:
2^8|a^2=e,b^2=h,d^2=g,e^2=f,f^2=h,b^a=bd,c^a=cg,d^a=df,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,e^c=eh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94774, #133261, #133262, … |
| maximal normal subgroup of | #94774, #133261, #133262, … |
| maximal quotient of | #124510, #124548, #124621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4904);