#6454041

order	512
nilpotency class	3
isoclinism	isoclinic class of #23900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

5041 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 16 points:

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

Polycyclic presentation:

2^9|e^2=g,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,f^c=fi,f^d=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361237);