Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,14,19,16,12,15,20)(11,21,24,27,26,22,25,17)(13,29,30,28,32,34,31,18)(23,33,37,36,39,40,38,35);(2,5)(6,7)(9,11,16,26)(10,17,12,27)(13,23,32,39)(14,25,15,24)(18,33,28,40)(19,22,20,21)(29,35,34,36)(30,38,31,37);(9,12,16,10)(11,22,26,21)(13,28,32,18)(14,19,15,20)(17,24,27,25)(23,36,39,35)(29,31,34,30)(33,38,40,37);(1,3,4,8)(2,6,5,7)(9,13)(10,18)(11,23)(12,28)(14,31)(15,30)(16,32)(17,33)(19,34)(20,29)(21,35)(22,36)(24,38)(25,37)(26,39)(27,40)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=ef,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh,e^d=eh,f^d=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #151233, #151235, #151409, … |
| maximal normal subgroup of | #151233, #151235, #151409, … |
| maximal quotient of | #312048, #312049, #312050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19913);