#168007

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
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

931 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)(11,20)(12,17)(13,18)(14,23)(15,19)(16,21)(22,24)(25,26)(27,35)(28,31)(29,33)(30,34)(32,38)(36,39)(37,40);(1,2,4,6)(3,5,7,8)(9,11,15,12)(10,16,19,24)(13,21,14,22)(17,23,20,18)(26,31)(28,29)(32,39)(33,34)(35,40)(37,38);(1,3)(2,5)(4,7)(6,8)(9,12,15,11)(10,17,19,20)(13,21,14,22)(16,23,24,18)(25,27,30,36)(26,32,34,40)(28,38,29,37)(31,39,33,35);(9,13,15,14)(10,18,19,23)(11,21,12,22)(16,17,24,20)(27,36)(32,40)(35,39)(37,38)

Polycyclic presentation:

2^9|b^2=gh,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, 75203);