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C₄²×D₈ #20497

order	256
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

C₄²×D₈

Subgroups

	subgroup	G/H
center	C₂×C₄²	D₄
commutator	C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂³	C₂²×D₄
prime radical	C₂²×D₈	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₈×C₄²⋊C₃
maximal subgroup of	C₄×#1331, #128531, C₄×#1333, …
maximal normal subgroup of	C₄×#1331, #128531, C₄×#1333, …
maximal quotient of	C₄×C₈×D₈, #183883, #183907, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13485);