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) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,15)(12,13)(14,16);(2,4)(9,11,13,16)(10,14,12,15);(5,6)(7,8)(11,16)(14,15);(1,2)(3,4)(9,11)(10,15)(12,14)(13,16);(9,12,13,10)(11,14,16,15)Polycyclic presentation:
2^8|a^2=g,b^2=h,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 | #132598, #132717, #487403, … |
| maximal normal subgroup of | #132598, #132717, #487403, … |
| maximal quotient of | #535288, #535923, #536492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38232);