#6629531

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
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

2196 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)(11,23)(12,17)(13,28)(14,20)(15,21)(16,22)(18,26)(19,29)(24,38)(25,33)(27,37)(30,35)(31,36)(32,39)(34,40);(2,5)(6,8)(9,11,14,12,16,26,15,25)(10,17,21,23,22,33,20,18)(13,24,29,27,30,39,31,40)(19,32,28,37,36,38,35,34);(1,3,4,7)(2,6,5,8)(9,12,14,26,16,25,15,11)(10,18,21,17,22,23,20,33)(13,27,29,39,30,40,31,24)(19,34,28,32,36,37,35,38);(9,13)(10,19)(11,24)(12,27)(14,29)(15,31)(16,30)(17,32)(18,34)(20,35)(21,28)(22,36)(23,37)(25,40)(26,39)(33,38);(9,14,16,15)(10,20,22,21)(11,12,26,25)(13,29,30,31)(17,18,33,23)(19,35,36,28)(24,27,39,40)(32,34,38,37)

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,c^b=ci,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536727);