#321736

order	512
nilpotency class	3
isoclinism	isoclinic class of #278235
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

975 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=eg,c^2=i,d^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228932);