#6435593

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433077
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60426	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 6342789);