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SD₁₆×#106 #225589

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

768 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);(1,2,3,4)(14,17)(15,19)(18,20);(13,14,15,17,16,18,19,20)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×#442, #72279283, #106×AGL(1,ℤ/8), …
maximal normal subgroup of	SD₁₆×#442, #72279283, #106×AGL(1,ℤ/8), …
maximal quotient of	#72309514, #72309665, SD₁₆×#365

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132785);