#513749

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂²	C₂³×C₄²
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

8698 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420945);