Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | Aut(G) = C24×#12067 (order 4096) · Inn(G) = C2×D16 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
149 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(6,12)(7,11)(8,18)(10,20)(13,28)(14,27)(15,30)(16,31)(17,23)(19,24)(21,33)(22,32)(25,26)(29,35)(34,36);(5,6)(7,13)(8,19)(9,14)(10,24)(11,25)(12,18)(15,20)(16,30)(17,29)(21,34)(22,35)(23,36)(26,32)(27,33)(28,31);(2,4)(5,7,9,11)(6,13,14,25)(8,16,21,22)(10,17,23,20)(12,26,27,28)(15,24,29,36)(18,32,33,31)(19,30,34,35)Polycyclic presentation:
2^8|c^2=h,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13663, #512826, #512829, … |
| maximal normal subgroup of | C2×#13663, #512826, #512829, … |
| maximal quotient of | #151172, #151284, #151482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6651);