Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9974 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
358 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 51 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 — 3 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,19)(11,20,24,32)(12,25,29,35)(13,30,14,18)(16,23,33,22)(17,27,36,28)(21,38,39,31)(26,37,40,34);(5,8)(9,11)(10,16)(12,26)(13,23)(14,22)(15,24)(17,34)(18,32)(19,33)(20,30)(21,27)(25,31)(28,39)(29,40)(35,38)(36,37);(1,3)(2,5)(4,7)(6,8)(9,12,15,29)(10,17,19,36)(11,21,24,39)(13,27,14,28)(16,31,33,38)(18,35,30,25)(20,37,32,34)(22,26,23,40)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fh,c^b=cf,d^b=dg,f^b=fg,d^c=dh,e^c=eg,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | #93206, #93208, #123021, … |
| maximal normal subgroup of | #93206, #93208, #123021, … |
| maximal quotient of | #102433, #102434, #102445, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2964);