#347567

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#15124	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1732 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,c^b=ch,f^c=fh,g^c=gi,f^d=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254763);