#7083556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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	#24372	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,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, 6990752);