#299662

order	512
nilpotency class	3
isoclinism	isoclinic class of #299653
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1137 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206858);