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Q₈×#106 #22724

order	256
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

Subgroups

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

419 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:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	SL(2,𝔽₃)×#106
maximal subgroup of	#136378, #136438, #136918, …
maximal normal subgroup of	#136378, #136438, #136918, …
maximal quotient of	Q₈×#334, #226370, #226382, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 15712);