Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 131072 |
| 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 | #2535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,7)(8,11,12,10)(13,14)(15,17)(16,19)(18,20);(5,6)(7,9)(8,11)(10,12)(13,15,18,16)(14,19,20,17);(1,2,3,4)(6,7)(10,11)(13,16,18,15)(14,19,20,17);(5,7,9,6)(8,11,12,10)(13,17,18,19)(14,15,20,16);(5,8,9,12)(6,10,7,11)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=fg,d^2=f,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 | #130703, #130869, #245721, … |
| maximal normal subgroup of | #130703, #130869, #245721, … |
| maximal quotient of | #535403, #535555, #536059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38325);