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SD₁₆×#97 #183992

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#97×AΓL(1,𝔽₉), #97×GL(2,𝔽₃), SD₁₆×#442, …
maximal normal subgroup of	SD₁₆×#442, #72279454, #97×AGL(1,ℤ/8), …
maximal quotient of	#97×AΓL(1,𝔽₉), SD₁₆×#347

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91188);