#6544936

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535696
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	#61066	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1945 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:

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

Polycyclic presentation:

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

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, 6452132);