#6574967

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1962 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482163);