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C₄×#1898 #354272

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#23900, AGL(1,𝔽₅)×#1898, C₄×#8534, …
maximal normal subgroup of	C₄×#23900, C₄×#8534, C₄×#23996, …
maximal quotient of	AGL(1,𝔽₅)×#1898, C₄×#9495, #72264005, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261468);