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C₄×#1715 #213101

order	512
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₈

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#127
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#1715, C₄×#19745, C₄×#12366
maximal normal subgroup of	C₄×#19745, C₄×#12366
maximal quotient of	AGL(1,𝔽₅)×#1715, C₄×#9837

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120297);