#6491813

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
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1935 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)(12,26)(13,20)(14,19)(15,18)(16,24)(17,28)(22,34)(23,33)(25,32)(27,31)(29,36)(30,35)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,24,15,23,14,25)(10,16,19,21,18,32,20,33)(12,27,28,39,29,22,30,38)(17,34,26,40,35,31,36,37);(2,5)(6,8)(9,12)(10,17)(11,22)(13,28)(14,30)(15,29)(16,31)(18,35)(19,26)(20,36)(21,37)(23,27)(24,38)(25,39)(32,34)(33,40);(2,5)(6,8)(11,23)(16,32)(21,33)(22,27)(24,25)(31,34)(37,40)(38,39);(1,3,4,7)(2,6,5,8)(10,18)(11,23)(17,35)(19,20)(22,27)(24,25)(26,36)(38,39)

Polycyclic presentation:

2^9|b^2=f,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,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, 6399009);