Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
1288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,9,8)(7,11,12,10)(13,14)(15,17)(16,19)(18,20);(2,4)(6,8)(10,11)(13,15,16,20)(14,17,19,18);(1,2)(3,4)(13,15)(14,18)(16,20)(17,19);(5,7,9,12)(6,10,8,11);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,14)(15,18)(16,19)(17,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6533892, #6533896, #6533902, … |
| maximal normal subgroup of | #6533892, #6533896, #6533902, … |
| maximal quotient of | #521326, #522070, #522586, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30657);