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Q₈×#97 #20522

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

Subgroups

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

477 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);(13,14,15,18,16,17,19,20);(1,2,3,4)(14,17)(18,20)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	SL(2,𝔽₃)×#97, #97×GL(2,𝔽₃)
maximal subgroup of	#100461, #100465, #100469, …
maximal normal subgroup of	#100461, #100465, #100469, …
maximal quotient of	Q₈×#322, #184389, #184402, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13510);