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C₂×#436 #2928

order	128
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

Subgroups

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

276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 14 points:

(1,2);(3,4)(5,6)(7,8,11,9,10,12,14,13);(4,6)(8,12)(9,13);(7,9,11,12,10,13,14,8)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#8166, #8206, #8208, …
maximal normal subgroup of	#8166, #8206, #8208, …
maximal quotient of	C₂×#1589, C₂×#1590, C₂×#1591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1660);