#6629527

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	#60415	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,17)(13,27)(14,20)(15,21)(16,22)(19,28)(23,31)(24,33)(25,32)(26,39)(29,36)(30,35)(34,37)(38,40);(2,5)(6,8)(9,11,14,24,16,25,15,12)(10,17,21,32,22,33,20,18)(13,23,28,37,30,38,29,26)(19,31,27,39,36,40,35,34);(1,3,4,7)(2,6,5,8)(9,12,15,25,16,24,14,11)(10,18,20,33,22,32,21,17)(13,26,29,38,30,37,28,23)(19,34,35,40,36,39,27,31);(9,13)(10,19)(11,23)(12,26)(14,28)(15,29)(16,30)(17,31)(18,34)(20,35)(21,27)(22,36)(24,37)(25,38)(32,39)(33,40);(9,14,16,15)(10,20,22,21)(11,24,25,12)(13,28,30,29)(17,18,33,32)(19,35,36,27)(23,37,38,26)(31,34,40,39)

Polycyclic presentation:

2^9|b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,h^a=hi,h^d=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536723);