Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
523 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 523 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,11,15,12,7,10,14)(8,9,19,17,13,16,18,20);(5,7,12,6)(8,17,13,20)(9,18,16,19)(10,14,11,15);(2,4)(5,8,12,13)(6,9,7,16)(10,18,11,19)(14,20,15,17);(2,4)(5,9)(6,13)(7,8)(10,17)(11,20)(12,16)(14,18)(15,19)Polycyclic presentation:
2^8|a^2=eh,b^2=h,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,g^b=gh,g^c=gh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #153139, #153140, #153145, … |
| maximal normal subgroup of | #153139, #153140, #153145, … |
| maximal quotient of | #332844, #332850, #332858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23477);