#6444366

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35634	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2807 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351562);