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C₄×#1335 #128525

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₄×C₈	D₈
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 ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

486 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);(5,6,9,14,11,15,20,13,12,16,8,19,10,7,17,18);(5,7)(6,11)(8,13)(9,18)(10,16)(12,15)(14,20)(17,19)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216211, #72216228, #72216233, …
maximal normal subgroup of	#72216211, #72216228, #72216233, …
maximal quotient of	#72216536, AGL(1,𝔽₅)×#1335, C₂×C₄×#1335, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35721);