Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62243 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1299 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1299 conjugacy classes of subgroups — too many to draw. Only its 487 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:
(2,5)(3,7)(10,13)(11,14)(15,16);(1,2)(3,6)(4,5)(7,8);(9,10,14,16,12,15,11,13);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|c^2=gh,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df64 conjugacy classes — view the full character table.
| Fitting subgroup of | #62415⋊C3 |
| maximal subgroup of | #131650, #131654, #250071, … |
| maximal normal subgroup of | #131650, #131654, #250071, … |
| maximal quotient of | #6386874, #6389484, #6389847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55403);