Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 366 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,7)(3,11)(4,5)(8,12)(9,16)(13,15);(1,2,4,14,6,7,5,10)(3,12,11,16,13,9,15,8);(1,3)(2,8,7,16)(4,15)(5,11)(6,13)(9,10,12,14);(1,3)(2,9)(4,15)(5,11)(6,13)(7,12)(8,10)(14,16)Polycyclic presentation:
2^8|b^2=ef,c^2=e,f^2=h,g^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,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6792410, #6792412, #6792416, … |
| maximal normal subgroup of | #6792410, #6792412, #6792416, … |
| maximal quotient of | #345968, #345970, #345973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24402);