Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
573 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 573 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,23)(13,18)(14,27)(15,19)(17,25)(20,36)(21,30)(22,32)(24,34)(26,35)(28,31)(29,37)(33,39)(38,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,20)(13,21)(14,28)(15,22)(17,33)(18,30)(19,32)(23,36)(24,37)(25,39)(26,38)(27,31)(29,34)(35,40);(9,12,15,26)(10,17,19,34)(11,20,22,38)(13,25,14,24)(16,29,32,33)(18,35,27,23)(21,39,28,37)(30,36,31,40);(9,13,15,14)(10,18,19,27)(11,21,22,28)(12,24,26,25)(16,30,32,31)(17,23,34,35)(20,37,38,39)(29,40,33,36)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142325, #142327, #142453, … |
| maximal normal subgroup of | #142325, #142327, #142453, … |
| maximal quotient of | #237746, #237749, #237753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16452);