Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | Aut(G) = #72231459 (order 2048) · Inn(G) = #442 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #442 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
99 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10,13,17,15,19,14,30)(11,20,23,38,21,39,24,35)(12,25,28,37,27,36,29,18)(16,31,32,22,33,26,34,40);(2,6)(3,7)(9,11,15,21)(10,16,19,33)(12,26,27,31)(13,24,14,23)(17,34,30,32)(18,35,37,38)(20,25,39,36)(22,29,40,28);(1,3,5,6,4,7,8,2)(10,17,19,30)(11,21)(12,27)(16,32,33,34)(18,36,37,25)(20,35,39,38)(23,24)(28,29)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=ef,d^2=f,e^2=g,f^2=h,b^a=bd,c^a=cfg,d^a=df,f^a=fh,c^b=cg,d^b=dfh,e^b=eh,f^b=fh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151120, C2×#12088, #483836, … |
| maximal normal subgroup of | #151120, C2×#12088, #483836, … |
| maximal quotient of | #128739, #129177, #135933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5076);