#166130

order	512
nilpotency class	3
isoclinism	isoclinic class of #165873
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

911 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 911 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,23,24,20)(25,26,29,32)(27,33,35,40)(28,36,38,31)(30,37,39,34);(2,5)(3,8)(6,7)(9,11,13,18)(10,14,16,21)(12,17,19,24)(15,20,22,23);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,24)(21,23)(25,27,29,35)(26,30,32,39)(28,37,38,34)(31,40,36,33);(2,6)(5,7)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 73326);