#6512733

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499844
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	#62830	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3140 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)(11,21,23,35)(12,25,27,36)(13,20,14,19)(16,29,32,24)(17,30,34,28)(22,31,26,33)(37,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,26)(13,29)(14,24)(15,23)(17,33)(18,32)(19,35)(20,21)(22,27)(25,37)(28,38)(30,39)(31,34)(36,40);(2,5)(3,7)(9,12)(10,17)(11,22)(13,28)(14,30)(15,27)(16,31)(18,34)(19,25)(20,36)(21,37)(23,26)(24,38)(29,39)(32,33)(35,40);(1,3,4,7)(2,6,5,8)(10,18)(12,27)(16,32)(19,20)(21,35)(22,26)(28,30)(38,39);(2,5)(6,8)(11,23)(16,32)(21,35)(22,26)(24,29)(31,33)(37,40)(38,39)

Polycyclic presentation:

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

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, 6419929);