#6542431

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62925	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2948 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:

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

Polycyclic presentation:

2^9|a^2=fg,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,e^d=eh

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, 6449627);