Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,8,10)(7,11,12,9)(14,18)(19,20);(2,4)(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,20)(17,18);(5,7)(6,9)(8,12)(10,11)(13,15,17,16)(14,19,18,20);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,16,17,15)(14,20,18,19);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fg,d^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 | C2×#59150, #7665748, #7666079, … |
| maximal normal subgroup of | C2×#59150, #7665748, #7666079, … |
| maximal quotient of | #531682, #531710, #531861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52138);