#6693554

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#33166	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gi,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6600750);