#6514393

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60565	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1701 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,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, 6421589);