#511790

order	512
nilpotency class	4
isoclinism	isoclinic class of #511790
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂	#32898
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3019 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 3019 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

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

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

Polycyclic presentation:

2^9|b^a=be,c^a=cf,d^a=dg,h^a=hi,f^b=fhi,g^b=gi,e^c=eh,e^d=ei,f^e=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418986);