#6494057

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473153
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2158 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6401253);