Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29751 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
445 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 445 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)(3,6)(4,5)(7,8)(9,10,13,19,15,18,14,20)(11,16,22,32,23,31,29,36)(12,24,26,34,28,35,27,17)(21,33,25,39,37,40,38,30);(2,5)(6,8)(9,11)(10,16)(12,25)(13,22)(14,29)(15,23)(17,33)(18,31)(19,32)(20,36)(21,27)(24,39)(26,37)(28,38)(30,35)(34,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,28)(10,17,18,34)(11,21,23,37)(13,27,14,26)(16,30,31,39)(19,35,20,24)(22,38,29,25)(32,40,36,33);(3,7)(6,8)(10,18)(16,31)(17,34)(19,20)(24,35)(30,39)(32,36)(33,40)Polycyclic presentation:
2^8|a^2=f,c^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,d^c=de,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #142261, #142263, #142397, … |
| maximal normal subgroup of | #142261, #142263, #142397, … |
| maximal quotient of | #329516, #329518, #329519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22740);