#6517125

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502819
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

2116 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=hi,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

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, 6424321);