Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 |
| prime radical | #2433 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(13,14)(15,16)(17,19)(18,20);(1,2,4,3)(5,6)(7,10)(8,11)(9,12)(13,14)(15,16)(17,19)(18,20);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,19)(17,20);(5,7)(6,10)(8,12)(9,11)(13,16,17,18)(14,15,19,20);(5,6)(7,9)(8,11)(10,12)(13,15)(14,16)(17,20)(18,19)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=fg,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129948, #253220, #253222, … |
| maximal normal subgroup of | #129948, #253220, #253222, … |
| maximal quotient of | #517702, #517706, #517747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28720);