#6444320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438880
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2759 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

(2,4)(5,6,9,15,10,14,16,11)(7,12,17,20,8,13,18,19);(6,11)(7,8)(9,16)(12,20)(13,19)(14,15);(1,2,3,4)(5,7,9,17,10,8,16,18)(6,12,15,20,14,13,11,19);(5,8,9,18,10,7,16,17)(6,13,15,19,14,12,11,20);(6,14)(11,15)(12,13)(19,20)

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=f,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,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, 6351516);