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C₄×#2134 #493020

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×Q₈	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1403 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);(5,6,9,14,10,15,7,12)(8,11,16,19,17,13,18,20);(6,11)(7,9)(8,17)(12,20)(13,15)(14,19);(5,7,10,9)(6,12,15,14)(8,16,17,18)(11,19,13,20)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281168, C₄×#12711, C₄×#33570, …
maximal normal subgroup of	#72281168, C₄×#12711, C₄×#33570, …
maximal quotient of	#72282604

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400216);