#6514280

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500482
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1725 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6421476);