Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
389 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 223 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)(9,10,14,18,15,19,13,29)(11,20,24,31,22,34,23,16)(12,25,27,32,28,39,26,17)(21,35,38,40,36,30,37,33);(2,5)(6,8)(9,11,13,23,15,22,14,24)(10,16,18,20,19,31,29,34)(12,21,26,37,28,36,27,38)(17,30,25,33,32,35,39,40);(1,3,4,7)(2,6,5,8)(9,12,13,26,15,28,14,27)(10,17,18,25,19,32,29,39)(11,21,23,37,22,36,24,38)(16,30,20,33,31,35,34,40);(11,22)(16,31)(20,34)(21,36)(23,24)(30,35)(33,40)(37,38)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #135400, #138380, #138381, … |
| maximal normal subgroup of | #135400, #138380, #138381, … |
| maximal quotient of | #191167, #191175, #191183, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13756);