#6519816

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500474
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1927 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)(11,21)(12,25)(13,18)(14,19)(15,20)(16,23)(17,29)(22,34)(24,32)(26,31)(27,36)(28,35)(30,33)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,26)(13,30)(14,23)(15,24)(17,34)(18,21)(19,32)(20,33)(22,27)(25,39)(28,38)(29,40)(31,35)(36,37);(2,5)(3,7)(9,12,14,29,15,27,13,28)(10,17,18,25,20,35,19,36)(11,22,23,38,24,26,30,40)(16,31,21,37,33,34,32,39);(1,3,4,7)(2,6,5,8)(12,27)(17,35)(22,26)(25,36)(28,29)(31,34)(37,39)(38,40);(2,5)(6,8)(9,13,15,14)(10,18,20,19)(11,23,24,30)(12,28,27,29)(16,32,33,21)(17,25,35,36)(22,38,26,40)(31,39,34,37)

Polycyclic presentation:

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

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, 6427012);