#505679

order	512
nilpotency class	4
isoclinism	isoclinic class of #500109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=e,d^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fhi,h^c=hi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412875);