#6435587

order	512
nilpotency class	3
isoclinism	isoclinic class of #29751
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

1906 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342783);