#7086569

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,32)(28,30)(29,31);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,21)(20,27)(24,31)(25,26)(28,29);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19,22,21)(18,23,26,25)(20,28,29,27)(24,32,31,30);(11,16)(13,15)(17,20,22,29)(18,24,26,31)(19,27,21,28)(23,30,25,32);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(17,19,22,21)(18,23,26,25)(20,27,29,28)(24,30,31,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=i,e^2=g,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, 6993765);