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M₁₆×#99 #183088

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

M₁₆×#99

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1465 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:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313059, #99×AGL(1,ℤ/8), M₁₆×B(4,𝔽₂), …
maximal normal subgroup of	#72313059, #99×AGL(1,ℤ/8), M₁₆×B(4,𝔽₂), …
maximal quotient of	#97×#99

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90284);