Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2688 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2688 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,18)(5,21)(6,24)(7,12)(8,16)(9,20)(10,23)(11,26)(14,29)(15,31)(17,27)(19,32)(22,30)(25,28);(1,3,6,25,7,16,26,17)(2,8,11,28,12,13,24,27)(4,19,20,30,10,14,21,31)(5,22,23,32,9,15,18,29);(1,4,7,10)(2,9,12,5)(3,14)(6,21,26,20)(8,15)(11,23,24,18)(13,22)(16,19)(17,31)(25,30)(27,29)(28,32);(1,5,7,9)(2,10,12,4)(3,15,16,22)(6,18,26,23)(8,19,13,14)(11,20,24,21)(17,29,25,32)(27,30,28,31)Polycyclic presentation:
2^8|b^2=ef,c^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6792404, #6792409, #6792444, … |
| maximal normal subgroup of | #6792404, #6792409, #6792444, … |
| maximal quotient of | #345973, #345974, #345977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24409);