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 | |
1237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,11)(9,12)(14,17)(15,19);(2,4)(5,6,10,11)(7,12,8,9)(14,17)(15,19);(5,7,10,8)(6,12,11,9)(13,14)(15,18)(16,17)(19,20);(1,2)(3,4)(5,8,10,7)(6,9,11,12)(13,15,16,19)(14,18,17,20);(5,9)(6,7)(8,11)(10,12)Polycyclic presentation:
2^8|b^2=f,c^2=f,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×#59092, #7666319, #7666410, … |
| maximal normal subgroup of | C2×#59092, #7666319, #7666410, … |
| maximal quotient of | #531634, #531704, #532095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52080);