#6513604

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1891 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)(12,24)(13,17)(14,19)(15,22)(16,26)(18,28)(21,29)(23,30)(25,33)(27,32)(31,38)(34,36)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,22,14,23,28,38)(10,15,18,20,19,31,17,30)(12,21,25,35,27,37,26,36)(16,29,32,34,33,40,24,39);(2,5)(3,7)(9,12)(10,16)(11,21)(13,25)(14,27)(15,29)(17,24)(18,32)(19,33)(20,34)(22,35)(23,37)(26,28)(30,39)(31,40)(36,38);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,18)(14,19)(17,28)(20,22)(21,29)(23,31)(24,26)(25,32)(27,33)(30,38)(34,35)(36,39)(37,40);(2,5)(6,8)(9,13,14,28)(10,17,19,18)(11,22,23,38)(12,25,27,26)(15,30,31,20)(16,24,33,32)(21,35,37,36)(29,39,40,34)

Polycyclic presentation:

2^9|b^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,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, 6420800);