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) = C29 |
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 | |
1975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(6,7)(9,10)(11,15)(12,16)(13,14);(2,4)(6,7)(10,14)(11,16);(5,6,8,7)(9,11)(10,12)(13,16)(14,15);(1,2)(3,4)(5,7,8,6)(9,11)(10,12)(13,16)(14,15);(9,12)(10,11)(13,15)(14,16)Polycyclic presentation:
2^8|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×#59123, #7665748, #7666078, … |
| maximal normal subgroup of | C2×#59123, #7665748, #7666078, … |
| maximal quotient of | #531682, #531709, #531763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52111);