#6586812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 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)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,24)(19,28,31,25)(20,22,27,30)(21,26,29,32);(2,5)(6,8)(17,19,23,31)(18,25,24,28)(20,32,27,26)(21,22,29,30);(10,13)(14,16)(17,20)(18,22)(19,29)(21,31)(23,27)(24,30)(25,26)(28,32);(9,11,12,15)(10,14,13,16)(17,21,23,29)(18,26,24,32)(19,30,31,22)(20,25,27,28);(1,3,4,7)(2,6,5,8)(17,22,23,30)(18,27,24,20)(19,26,31,32)(21,28,29,25)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6494008);