#6617823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574100
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	#61403	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1675 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6525019);