Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
2205 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 2205 conjugacy classes of subgroups — too many to draw. Only its 503 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,4,3)(6,8)(9,10)(11,15)(12,13)(14,16);(2,3)(9,10)(11,15)(12,13)(14,16);(5,6,7,8)(10,13)(14,15);(1,3,4,2)(9,10)(11,14)(12,13)(15,16);(9,11,12,16)(10,14,13,15)Polycyclic presentation:
2^8|a^2=f,c^2=gh,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 | #130752, #132561, #245711, … |
| maximal normal subgroup of | #130752, #132561, #245711, … |
| maximal quotient of | #535411, #536068, #536714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38343);