Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
1251 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,7)(10,11)(13,14)(15,19)(16,17)(18,20);(6,7)(8,12)(14,16)(18,19);(1,2)(3,4)(5,6,9,7)(8,10,12,11)(13,14,17,16)(15,18,20,19);(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,19)(17,20);(5,8)(6,10)(7,11)(9,12)(13,16,17,14)(15,19,20,18)Polycyclic presentation:
2^8|c^2=h,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97318, #246757, #246764, … |
| maximal normal subgroup of | #97318, #246757, #246764, … |
| maximal quotient of | #557049, #557051, #557362, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39212);