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C₄×#2238 #495939

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄	#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#481
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

625 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);(6,10)(7,13)(8,16)(11,18)(12,15)(14,20)(17,19);(7,14)(10,18)(13,20)(15,19);(5,6,8,17,9,11,16,12)(7,15,13,18,14,19,20,10)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#33688, AGL(1,𝔽₅)×#2238
maximal normal subgroup of	C₄×#33688
maximal quotient of	C₄×#13138, AGL(1,𝔽₅)×#2238

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403135);