Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61109 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1342 conjugacy classes of subgroups; only the 485 normal subgroups are shown.
This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 485 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 — 5 generators on 16 points:
(1,2,7,10)(3,12,8,15)(4,14,5,9)(6,13,16,11);(1,3)(2,5)(4,10)(6,9)(7,8)(11,15)(12,13)(14,16);(1,4)(2,8)(3,10)(5,7)(6,15)(9,11)(12,16)(13,14);(1,5)(2,9)(3,13)(4,7)(6,15)(8,11)(10,14)(12,16);(2,10)(3,8)(9,14)(11,13)Polycyclic presentation:
2^8|a^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gh,d^b=df,e^b=eh,f^b=fh,g^c=gh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152180, #152191, #152192, … |
| maximal normal subgroup of | #152180, #152191, #152192, … |
| maximal quotient of | #6358843, #6358848, #6358850, … |
| holomorph of | SD16 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54098);