Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | Aut(G) = #72216906 (order 1024) · Inn(G) = #357 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #357 |
| commutator | C16 | C2×C8 |
| Frattini, cosocle | C4×C16 | C22 |
| socle | C2 | #1331 |
| prime radical | #508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
59 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,12,29,39,8,14,24,44,32,13,6,30)(3,15,17,47,19,49,31,48,21,50,18,56,20,57,52,38)(4,22,25,40,27,58,51,42,28,46,62,41,63,59,26,10)(9,33,34,54,36,61,35,16,37,23,43,53,45,60,64,55);(1,3)(2,9)(4,23)(5,18)(6,31)(7,19)(8,21)(10,38)(11,43)(12,36)(13,45)(14,37)(15,46)(16,51)(17,24)(20,32)(22,50)(25,54)(26,55)(27,60)(28,33)(29,52)(30,35)(34,44)(39,64)(40,47)(41,56)(42,48)(49,59)(53,62)(57,58)(61,63)Polycyclic presentation:
2^8|a^2=d,c^2=egh,d^2=f,e^2=gh,f^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ceh,d^b=dh,e^b=eg,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7388, #150069, #150144, … |
| maximal normal subgroup of | C2×#7388, #150069, #150144, … |
| maximal quotient of | #93749, #93750, #93762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 376);