Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
477 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 477 conjugacy classes of subgroups — too many to draw. Only its 183 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:
(2,5)(3,7)(9,10,14,18,13,17,27,34)(11,19,22,36,21,15,29,31)(12,23,26,35,25,16,28,33)(20,30,37,39,24,32,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,24)(13,21)(14,22)(16,32)(17,19)(18,31)(20,25)(23,30)(26,38)(27,29)(28,37)(33,40)(34,36)(35,39);(2,5)(6,8)(9,12,13,25)(10,16,17,23)(11,20,21,24)(14,28,27,26)(15,30,19,32)(18,35,34,33)(22,38,29,37)(31,40,36,39);(1,3,4,7)(2,6,5,8)(10,17)(11,21)(16,23)(18,34)(20,24)(22,29)(33,35)(37,38)Polycyclic presentation:
2^8|a^2=g,c^2=h,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #214929, #214931, #214933, … |
| maximal normal subgroup of | #214929, #214931, #214933, … |
| maximal quotient of | #216195, #216197, #216201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14871);