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C₄×#366 #12672

order	256
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₈
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle, prime radical	C₂³	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

183 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94044, #94051, #94409, …
maximal normal subgroup of	#94044, #94051, #94409, …
maximal quotient of	C₄×#1371, #129112, #129129, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5660);