#472559

order	512
nilpotency class	3
isoclinism	isoclinic class of #457592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

877 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egh,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=hi,f^d=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379755);