#403526

order	512
nilpotency class	3
isoclinism	isoclinic class of #393934
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	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1944 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)(17,20);(1,2,4,3)(5,6,8,10)(7,11,12,9);(5,7,8,12)(6,9,10,11)(13,14)(15,17)(16,18)(19,20);(1,3,4,2)(13,15)(14,17)(16,19)(18,20)

Polycyclic presentation:

2^9|b^2=eg,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^c=hi,f^d=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310722);