#168006

order	512
nilpotency class	3
isoclinism	isoclinic class of #167761
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

757 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,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, 75202);