C₁₆×#131 #479940

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₆ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₁₆×#131

Subgroups

	subgroup	G/H
center	C₈×C₁₆	C₂²
commutator	C₂	C₂²×C₄×C₁₆
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₄×C₈
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16);(3,9)(6,12)(10,15)(13,16)

Conjugacy classes & character table

320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	M₃₂×#131, C₁₆×#504, #72226970, …
maximal normal subgroup of	M₃₂×#131, C₁₆×#504, #72226970, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387136);

C16×#131 #479940