Finite Group Databasebeta

← Prev (#497393) Next (#497395) →

D₄×#361 #497394

order	512
nilpotency class	4
isoclinism	isoclinic class of #481415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×D₄×D₈	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227033, D₄×AGL(1,ℤ/16), #72227163, …
maximal normal subgroup of	#72227033, D₄×AGL(1,ℤ/16), #72227163, …
maximal quotient of	D₄×#1333, #72227031, #72227033, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404590);