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C₁₆×AGL(1,ℤ/8) #483618

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

C₁₆×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₄	C₂³×C₁₆
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×AGL(1,ℤ/8)	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(1,2)(3,5)(4,6)(7,8);(2,5)(3,8)(6,7);(2,6)(5,7)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×AGL(1,𝔽₁₇), C₁₆×#453, M₃₂×AGL(1,ℤ/8), …
maximal normal subgroup of	C₁₆×#453, M₃₂×AGL(1,ℤ/8), #72320682, …
maximal quotient of	AGL(1,ℤ/8)×AGL(1,𝔽₁₇), #72287273

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390814);