Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
1809 conjugacy classes of subgroups; only the 595 normal subgroups are shown.
This group has 1809 conjugacy classes of subgroups — too many to draw. Only its 595 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) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(6,7)(9,10,13,15)(11,12,14,16);(5,6,8,7)(9,11)(10,12)(13,14)(15,16);(1,2,3,4)(5,7,8,6)(9,12)(10,14)(11,15)(13,16);(9,11)(10,12)(13,14)(15,16);(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #36179⋊C3 |
| maximal subgroup of | #98246, #130621, #223156, … |
| maximal normal subgroup of | #98246, #130621, #223156, … |
| maximal quotient of | #519051, #519377, #519380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29167);