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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,10)(11,12)(13,14,16,18)(15,19,20,17);(2,4)(5,6)(7,11)(8,12)(9,10)(15,20)(17,19);(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,17,18,19);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(13,14)(15,17)(16,18)(19,20);(5,6)(7,11)(8,12)(9,10)Polycyclic presentation:
2^8|a^2=g,c^2=fg,d^2=f,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×#59048, #7666696, #7666782, … |
| maximal normal subgroup of | C2×#59048, #7666696, #7666782, … |
| maximal quotient of | #531670, #531711, #532027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52036);