#6456440

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438880
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

1797 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)(9,10,14,19,15,18,26,34)(11,20,23,31,24,35,29,16)(12,25,27,40,13,17,28,33)(21,37,38,30,22,36,39,32);(9,11,15,24)(10,16,18,31)(12,22,13,21)(14,29,26,23)(17,32,25,30)(19,35,34,20)(27,38,28,39)(33,36,40,37);(2,5)(6,8)(9,12,14,27,15,13,26,28)(10,17,19,33,18,25,34,40)(11,21,23,38,24,22,29,39)(16,30,20,36,31,32,35,37);(9,13,14,28,15,12,26,27)(10,17,19,33,18,25,34,40)(11,22,23,39,24,21,29,38)(16,30,20,36,31,32,35,37);(1,3,4,7)(2,6,5,8)(10,18)(16,31)(17,25)(19,34)(20,35)(30,32)(33,40)(36,37)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=f,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,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, 6363636);