#413338

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#393
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1325 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)(7,14,16,18)(11,19,20,15);(5,6)(7,15,16,19)(8,17)(9,12)(10,13)(11,14,20,18);(5,7,8,18,10,16,9,14)(6,11,12,15,13,20,17,19);(1,2,3,4)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei,f^c=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320534);