#7083470

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	#42977	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(18,24)(21,22)(25,32)(26,30)(27,28)(29,31);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,27,23,28,22,24)(19,25,31,26,20,30,29,32);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,19,23,20)(18,25,28,30)(21,31,22,29)(24,32,27,26);(1,4,5,3)(2,7,8,6)(11,16)(13,15)(17,20,23,19)(18,26,28,32)(21,31,22,29)(24,30,27,25);(10,14)(13,15)(17,21,23,22)(18,27,28,24)(19,29,20,31)(25,32,30,26)

Polycyclic presentation:

2^9|b^2=fgi,c^2=ghi,d^2=gi,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, 6990666);