C₄×#1884 #354234

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₄×#1884

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1980 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)(7,14,15,18)(8,11,16,17)(9,12)(10,13)(19,20);(5,7,9,17,10,15,19,11)(6,8,12,18,13,16,20,14);(5,8)(6,11)(7,12)(9,18)(10,16)(13,17)(14,19)(15,20)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213889, #72214887, #72215866, …
maximal normal subgroup of	#72213889, #72214887, #72215866, …
maximal quotient of	AGL(1,𝔽₅)×#1884, #72303522

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261430);

C4×#1884 #354234

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₄×#1884 #354234