#6502946

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502937
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	#60456	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1914 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,d^2=i,e^2=i,f^2=i,g^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, 6410142);