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Q₈×#442 #6385102

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2287 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×#442, #72279433, #41×#442, …
maximal normal subgroup of	SD₁₆×#442, #72279433, #41×#442, …
maximal quotient of	#72309671, #72313619, Q₈×AGL(1,ℤ/16)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6292298);