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SD₁₆×#117 #6380874

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305393, SD₁₆×#525, #117×AGL(1,ℤ/8)
maximal normal subgroup of	#72305393, SD₁₆×#525, #117×AGL(1,ℤ/8)
maximal quotient of	#72305898

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288070);