Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
440 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 440 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,11)(7,14)(8,9)(12,15)(13,17)(19,20);(1,2)(3,4)(5,6)(7,15)(8,17)(9,11)(10,13)(12,16)(14,19)(18,20);(2,4)(5,6)(7,12)(8,11)(9,17)(10,13)(14,20)(15,16)(18,19);(5,7,8,16,10,14,9,18)(6,12,11,15,13,20,17,19)Polycyclic presentation:
2^8|d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,e^b=eh,f^c=fh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #151257, #151259, #151425, … |
| maximal normal subgroup of | #151257, #151259, #151425, … |
| maximal quotient of | #339703, #339705, #339715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23853);