Finite Group Databasebeta

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C₂×C₄×C₅₁₂ #72183627

order	4096
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂²×C₄

Aliases

abelian	C₂×C₄×C₅₁₂
multiplicative	(ℤ/10240)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₂₅₆	C₂³
socle, prime radical	C₂³	C₂×C₂₅₆
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₂×C₂₅₆ ▷ C₁₂₈ ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

243 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

4096 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₄×C₁₀₂₄, C₄²×C₅₁₂, C₂×C₈×C₅₁₂, …
maximal normal subgroup of	C₂×C₄×C₁₀₂₄, C₄²×C₅₁₂, C₂×C₈×C₅₁₂, …
maximal quotient of	C₂×C₄×C₁₀₂₄, C₄²×C₅₁₂, C₂×C₈×C₅₁₂, …
automorphism group of	C₅×C₂₀₄₈
outer automorphism group of	C₅×C₂₀₄₈

GAP

Past the SmallGroups library coverage — no SmallGroups id.