Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) = #72231459 (order 2048) · Inn(G) = #442 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #442 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,13,19,15,20,14,17)(11,21,23,38,25,39,24,35)(12,26,29,36,28,34,30,18)(16,31,32,22,33,27,37,40);(9,11,15,25)(10,16,20,33)(12,27,28,31)(13,24,14,23)(17,32,19,37)(18,35,36,38)(21,26,39,34)(22,30,40,29);(1,3,4,8,5,2,6,7)(10,17,20,19)(12,28)(16,32,33,37)(18,26,36,34)(21,35,39,38)(22,40)(27,31)(29,30)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=cf,d^a=df,f^a=fh,d^b=dfh,e^b=eh,f^b=fh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #151105, C2×#12066, #483589, … |
| maximal normal subgroup of | #151105, C2×#12066, #483589, … |
| maximal quotient of | #122962, #122965, #122969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5054);