#288715

order	512
nilpotency class	3
isoclinism	isoclinic class of #288715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25384	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

840 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195911);