Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6,9,8)(7,11,12,10)(13,14,17,15)(16,18,20,19);(1,2,4,3)(7,12)(10,11)(16,20)(18,19);(5,7)(6,10)(8,11)(9,12)(13,15,17,14)(16,19,20,18);(5,7)(6,10)(8,11)(9,12)(13,16)(14,18)(15,19)(17,20);(1,3,4,2)(5,8,9,6)(7,11,12,10)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #7006174, #7006186, #7006243, … |
| maximal normal subgroup of | #7006174, #7006186, #7006243, … |
| maximal quotient of | #531791, #531852, #532141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52743);