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D₅₂₀ #11760142

order	1040
automorphism	Aut(G) order 99840
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

D₅₂₀

Splittings

Fitting	C₅×C₈×C₁₃ ⋊ C₂
Hall	C₅×C₁₃ ⋊ D₈
Sylow	C₁₃ ⋊ D₄₀
Sylow	C₅ ⋊ D₁₀₄

Subgroups

	subgroup	G/H
center	C₂	D₂₆₀
commutator, cosocle	C₄×C₅×C₁₃	C₂²
Frattini	C₄	C₂×D₆₅
Fitting	C₅×C₈×C₁₃	C₂
socle	C₂×C₅×C₁₃	D₄
hypercenter	C₈	D₆₅
hypocenter	C₅×C₁₃	D₈
LCS	G ▷ C₄×C₅×C₁₃ ▷ C₂×C₅×C₁₃ ▷ C₅×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 520

A minimal faithful permutation representation — 2 generators on 26 points:

(1,2,3,4,5,6,7,8)(9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26);(2,8)(3,7)(4,6)(10,13)(11,12)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)

Polycyclic presentation:

2^4,5,13|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^12

Conjugacy classes & character table

263 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₀₄₀, D₁₅₆₀, D₂₆₀₀, …
maximal normal subgroup of	D₁₀₄₀
maximal quotient of	D₁₀₄₀, D₁₅₆₀, Dic₅₂₀, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1040, 71);