#7083468

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#28584	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=fgi,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,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, 6990664);