#6543971

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#26619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1514 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451167);