Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | Aut(G) = #72340709 (order 4096) · Inn(G) = #2669 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2669 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 113 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:
(1,2,5,8,7,10,6,9)(3,11,12,15,14,16,13,4);(1,2)(3,11)(4,12)(5,9)(6,8)(7,10)(13,15)(14,16);(1,3)(2,4)(5,12)(6,13)(7,14)(8,11)(9,16)(10,15);(1,4)(2,3)(5,16)(6,11)(7,15)(8,13)(9,12)(10,14)Polycyclic presentation:
2^8|a^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fh,g^c=gh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153147, #153149, #153151, … |
| maximal normal subgroup of | #153147, #153149, #153151, … |
| maximal quotient of | #332839, #332844, #332846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23484);