Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| 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 | |
505 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 505 conjugacy classes of subgroups — too many to draw. Only its 191 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)(9,10,13,18,14,17,19,31)(11,15,22,27,21,28,25,32)(12,16,24,30,23,29,33,39)(20,26,35,38,34,37,36,40);(9,11,19,22,14,21,13,25)(10,15,18,32,17,28,31,27)(12,20,33,35,23,34,24,36)(16,26,30,40,29,37,39,38);(2,5)(6,8)(9,12,14,23)(10,16,17,29)(11,20,21,34)(13,24,19,33)(15,26,28,37)(18,30,31,39)(22,35,25,36)(27,38,32,40);(1,3,4,7)(2,6,5,8)(12,23)(16,29)(20,34)(24,33)(26,37)(30,39)(35,36)(38,40)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=h,d^2=f,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #215091, #215095, #215099, … |
| maximal normal subgroup of | #215091, #215095, #215099, … |
| maximal quotient of | #207675, #207677, #207693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14417);