Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) order 1179648 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
2853 conjugacy classes of subgroups; only the 573 normal subgroups are shown.
This group has 2853 conjugacy classes of subgroups — too many to draw. Only its 573 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(5,6,7,8);(5,6)(7,8)(9,10,14,13)(11,16,12,15);(1,2)(3,4)(5,6)(7,8)(9,11,14,12)(10,15,13,16);(9,12,14,11)(10,15,13,16);(9,13,14,10)(11,16,12,15)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=h,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #11709949, #11709950, #11709951 |
| maximal subgroup of | #132213, #248653, #248665, … |
| maximal normal subgroup of | #132213, #248653, #248665, … |
| maximal quotient of | #517613, #517617, #517622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29079);