#6444416

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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

2777 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)(9,10,14,18,15,19,24,35)(11,20,22,30,23,34,12,16)(13,17,28,33,27,32,36,39)(21,31,26,38,37,40,25,29);(5,6)(7,8)(9,11,15,23)(10,16,19,30)(12,24,22,14)(13,26,27,25)(17,31,32,40)(18,34,35,20)(21,36,37,28)(29,39,38,33);(1,2,3,4)(6,8)(9,12,14,11,15,22,24,23)(10,16,18,20,19,30,35,34)(13,25,28,21,27,26,36,37)(17,29,33,31,32,38,39,40);(6,8)(9,13,14,28,15,27,24,36)(10,17,18,33,19,32,35,39)(11,21,22,26,23,37,12,25)(16,29,20,31,30,38,34,40);(13,27)(17,32)(21,37)(25,26)(28,36)(29,38)(31,40)(33,39)

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^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351612);