#7176672

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993915
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7083868);