#6491465

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
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	#62850	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2852 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:

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

Polycyclic presentation:

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

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, 6398661);