#6445667

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437793
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
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 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:

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

Polycyclic presentation:

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

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, 6352863);