#6454803

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 16 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15);(9,10,12,16,13,15,11,14);(2,5)(6,8)(9,10)(11,14)(12,16)(13,15);(9,10)(11,14)(12,16)(13,15);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226040, #72226052, #72226215, …
maximal normal subgroup of	#72226040, #72226215, #72226238
maximal quotient of	#72226934

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361999);