#6439374

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2042 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)(11,22)(12,26)(13,18)(14,19)(15,21)(16,23)(17,29)(20,32)(24,36)(25,38)(27,34)(28,35)(30,37)(31,40)(33,39);(2,5)(6,8)(9,11)(10,16)(12,27)(13,29)(14,23)(15,24)(17,37)(18,39)(19,22)(20,36)(21,38)(25,32)(26,40)(28,31)(30,33)(34,35);(9,12,14,28)(10,17,19,39)(11,13,23,30)(15,33,32,29)(16,34,22,40)(18,38,37,36)(20,35,21,26)(24,27,25,31);(1,3,4,7)(2,6,5,8)(9,13,14,30)(10,18,19,37)(11,12,23,28)(15,31,32,27)(16,35,22,26)(17,38,39,36)(20,34,21,40)(24,29,25,33);(10,19)(16,22)(17,39)(18,37)(20,21)(26,35)(34,40)(36,38)

Polycyclic presentation:

2^9|c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,h^c=hi,h^d=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, 6346570);