#6491812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470415
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

1924 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:

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

Polycyclic presentation:

2^9|b^2=f,d^2=i,e^2=h,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,e^b=ei,g^c=gi

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, 6399008);