Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
737 conjugacy classes of subgroups; only the 495 normal subgroups are shown.
This group has 737 conjugacy classes of subgroups — too many to draw. Only its 495 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(9,10,13,12)(11,15,16,14)(20,28)(24,29)(26,32)(30,31);(1,2,5,3)(4,7,8,6)(9,11,13,16)(10,14,12,15)(18,22)(20,28)(23,25)(26,32);(1,3,5,2)(4,7,8,6)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,14,16,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=g,d^2=fh,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 | #7151935, #7151939, #7151951, … |
| maximal normal subgroup of | #7151935, #7151939, #7151951, … |
| maximal quotient of | #535463, #536135, #536796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38415);