Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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 | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
447 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 447 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:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18,15,19,13,28)(11,16,23,31,21,30,22,37)(12,24,26,32,27,38,25,17)(20,33,36,39,34,40,35,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,29)(18,31)(19,30)(24,33)(25,35)(26,36)(27,34)(28,37)(32,39)(38,40);(3,8)(5,7)(9,12,13,25,15,27,14,26)(10,17,18,24,19,32,28,38)(11,20,22,35,21,34,23,36)(16,29,31,33,30,39,37,40);(2,6)(5,7)(11,21)(16,30)(20,34)(22,23)(29,39)(31,37)(33,40)(35,36)Polycyclic presentation:
2^8|a^2=g,c^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #135989, #135995, #136011, … |
| maximal normal subgroup of | #135989, #135995, #136011, … |
| maximal quotient of | #217278, #217279, #217280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14938);