Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,10)(7,8,11,12);(6,10)(7,11)(13,14)(15,17)(16,19)(18,20);(1,2)(3,4)(14,15)(18,19);(5,7)(6,8)(9,11)(10,12)(13,15,17,14)(16,19,20,18);(5,8)(6,7)(9,12)(10,11)(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^8|a^2=f,d^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133409, #499809, #500230, … |
| maximal normal subgroup of | #133409, #499809, #500230, … |
| maximal quotient of | #560391, #561490, #562693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43623);