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C₄×#1336 #128527

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₄²	#102
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×D₄⋊C₈
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2,3,4);(5,6,9,13,10,14,20,19,11,15,8,17,18,7,16,12);(5,7)(6,10)(8,12)(9,19)(11,14)(13,16)(15,18)(17,20)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216223, #72216228, #72216234, …
maximal normal subgroup of	#72216223, #72216228, #72216234, …
maximal quotient of	#72216529, AGL(1,𝔽₅)×#1336, C₈×#1336, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35723);