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D₄×#344 #356495

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2)(3,4);(2,4);(5,6)(7,13,14,10)(8,12)(9,11)(15,19,20,17)(16,18);(5,7,8,15,9,14,16,20)(6,10,12,17,11,13,18,19)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#1904, D₄×#2008, D₄×#1891, …
maximal normal subgroup of	D₄×#1904, D₄×#2008, D₄×#1891, …
maximal quotient of	#72343003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263691);