C₄×#1906 #354236

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₄×#1906

Subgroups

	subgroup	G/H
center	C₄²	#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2006 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);(5,6)(7,10,14,20)(8,13)(9,12)(11,15,19,16)(17,18);(5,7)(6,10,12,20)(8,16)(9,14)(11,18,19,13)(15,17);(5,7)(6,11)(8,16)(9,14)(10,18)(12,19)(13,20)(15,17)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#1906, C₄×#23872, #72224693, …
maximal normal subgroup of	C₄×#23872, #72224693, C₄×#24472
maximal quotient of	AGL(1,𝔽₅)×#1906, #72283346

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261432);

C4×#1906 #354236

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₄×#1906 #354236