#6574968

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

1983 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)(11,17)(12,26)(13,20)(14,18)(15,21)(16,22)(19,24)(23,32)(25,33)(27,36)(28,40)(29,34)(30,38)(31,37)(35,39);(2,5)(6,8)(9,11,16,25)(10,17,22,33)(12,14,24,15)(13,27,30,39)(18,19,21,26)(20,36,38,35)(23,31,28,29)(32,37,40,34);(1,3,4,7)(2,6,5,8)(10,18)(11,12)(14,15)(19,26)(20,34)(21,22)(23,27)(24,25)(28,39)(29,31)(35,36)(37,38);(9,12)(10,19)(11,14)(13,28)(15,25)(16,24)(17,21)(18,33)(20,32)(22,26)(23,30)(27,29)(31,39)(34,35)(36,37)(38,40);(9,13,16,30)(10,20,22,38)(11,23,25,28)(12,27,24,39)(14,29,15,31)(17,32,33,40)(18,34,21,37)(19,35,26,36)

Polycyclic presentation:

2^9|b^2=i,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, 6482164);