#6444751

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	#34925	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2406 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351947);