#6636794

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1525 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efhi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6543990);