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C₄×#1955 #354534

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

Subgroups

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

2434 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);(6,8)(10,12)(13,14,18,19,17,15,16,20);(5,6,7,8)(13,15,18,20,17,14,16,19);(9,10,11,12)(13,16,17,18)(14,19,15,20)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72211651, C₂×C₄×#1955, #72216201, …
maximal normal subgroup of	#72211651, C₂×C₄×#1955, #72216201, …
maximal quotient of	#72211620, C₂×C₄×#1955

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261730);