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C₄×#2122 #492992

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂²×D₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1427 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)(7,11)(8,15,16,10)(9,12)(13,19)(14,18,20,17);(6,10)(7,13)(11,18)(12,15)(14,20)(17,19);(5,7,9,13)(6,11,12,19)(8,14,16,20)(10,17,15,18)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#12718, C₄×#12755, C₄×#33553, …
maximal normal subgroup of	C₄×#12718, C₄×#12755, C₄×#33553, …
maximal quotient of	#72282727, C₂×C₄×#2122

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400188);