#6449943

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

1886 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)(3,6)(4,5)(7,8)(9,10,14,31,16,20,15,19)(11,21,32,17,25,34,24,35)(12,26,28,37,13,18,29,38)(22,36,27,40,23,39,30,33);(2,5)(6,8)(9,11)(10,17)(12,27)(13,30)(14,24)(15,32)(16,25)(18,36)(19,34)(20,35)(21,31)(22,28)(23,29)(26,39)(33,38)(37,40);(2,5)(6,8)(9,12,16,13)(10,18,20,26)(11,22,25,23)(14,28,15,29)(17,33,35,40)(19,37,31,38)(21,39,34,36)(24,30,32,27);(1,3,4,7)(2,6,5,8)(9,13,16,12)(10,18,20,26)(11,23,25,22)(14,29,15,28)(17,33,35,40)(19,37,31,38)(21,39,34,36)(24,27,32,30);(9,14,16,15)(10,19,20,31)(11,24,25,32)(12,28,13,29)(17,34,35,21)(18,37,26,38)(22,30,23,27)(33,36,40,39)

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=g,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, 6357139);