#41×#126 #7630346

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

#41×#126

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2)(3,6)(4,5)(7,8);(2,5)(6,8);(1,3,4,7)(2,6,5,8);(10,14)(11,18)(16,23)(17,21)(19,22)(20,24);(9,10,15,21)(11,16,22,23)(12,14,13,17)(18,24,19,20);(9,11,12,19)(10,16,14,20)(13,18,15,22)(17,24,21,23)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#41×#126⋊C₃
maximal subgroup of	#41×#126⋊C₃
maximal normal subgroup of	#41×#126⋊C₃
maximal quotient of	#126²

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537542);