Finite Group Databasebeta

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C₂²×C₄×C₁₂₈ #72182037

order	2048
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 3145728
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

abelian	C₂²×C₄×C₁₂₈
multiplicative	(ℤ/7680)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₆₄	C₂⁴
socle, prime radical	C₂⁴	C₂×C₆₄
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₂×C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

904 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

2048 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×C₄×C₂₅₆, C₂²×C₈×C₁₂₈, C₂×C₄²×C₁₂₈, …
maximal normal subgroup of	C₂²×C₄×C₂₅₆, C₂²×C₈×C₁₂₈, C₂×C₄²×C₁₂₈, …
maximal quotient of	C₂²×C₄×C₂₅₆, C₂²×C₈×C₁₂₈, C₂×C₄²×C₁₂₈, …
automorphism group of	C₃×C₅×C₅₁₂
outer automorphism group of	C₃×C₅×C₅₁₂

GAP

Past the SmallGroups library coverage — no SmallGroups id.