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D₄×#360 #497391

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	D₄×#135	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

1148 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);(6,10,11,19)(7,13)(8,16)(12,14,17,20)(15,18);(5,6)(7,14)(8,17)(9,11)(10,18)(12,16)(13,19)(15,20)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227032, D₄×AGL(1,ℤ/16), #72227162, …
maximal normal subgroup of	#72227032, D₄×AGL(1,ℤ/16), #72227162, …
maximal quotient of	D₄×#1333, #72227030, #72227032, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404587);