#167860

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=h,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, 75056);