#7101203

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	#31068	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=g,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,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, 7008399);