#267223

order	512
nilpotency class	3
isoclinism	isoclinic class of #267211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25721	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1378 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,f^c=fh,g^c=gi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174419);