Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(6,8)(10,11)(15,19)(18,20);(2,3)(6,8)(10,11)(13,14,17,16)(15,18,19,20);(5,6,9,8)(7,11,12,10)(13,15,17,19)(14,18,16,20);(1,3,4,2)(5,7,9,12)(6,10,8,11)(13,16,17,14)(15,20,19,18);(5,8,9,6)(7,11,12,10)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=g,d^2=fg,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7117002, #7117008, #7117018, … |
| maximal normal subgroup of | #7117002, #7117008, #7117018, … |
| maximal quotient of | #535408, #535554, #536063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38301);