#6516755

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500002
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62836	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3306 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)(11,20,23,31)(12,24)(13,27)(14,18)(15,19)(16,22,32,28)(17,26)(21,35,38,40)(25,33)(29,34)(30,36,39,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,30)(18,31)(19,32)(20,27)(24,39)(25,36)(26,38)(29,37)(33,40)(34,35);(2,5)(3,7)(9,12)(10,17)(11,21)(13,25)(14,29)(15,26)(16,30)(18,33)(19,24)(20,35)(22,36)(23,38)(27,34)(28,37)(31,40)(32,39);(1,3,4,7)(2,6,5,8);(2,5)(6,8)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,25,26,29)(16,31,32,20)(17,33,24,34)(21,36,38,37)(30,40,39,35)

Polycyclic presentation:

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

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, 6423951);