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C₂×C₈×M₁₆ #17312

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

C₂×C₈×M₁₆

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂²×C₄×C₈
Frattini, cosocle	C₄²	C₂⁴
socle	C₂³	C₂×C₄²
prime radical	C₂⁴	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 18 points:

(1,2);(3,4,5,6,7,8,9,10);(11,12,13,14,15,16,17,18);(12,16)(14,18)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×H(3,ℤ/4), #105262, C₈×#338, …
maximal normal subgroup of	C₈×H(3,ℤ/4), #105262, C₈×#338, …
maximal quotient of	C₂×C₈×#97, C₄×C₈×M₁₆, C₂×#7848, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10300);