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C₂×#428 #2910

order	128
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

Subgroups

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

268 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)(8,11)(9,13)(12,14);(7,8,9,11,10,12,13,14);(8,12)(11,14)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10896, #10899, #10903, …
maximal normal subgroup of	#10896, #10899, #10903, …
maximal quotient of	C₂×#1565, C₂×#1569, #17796, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1642);