#6679564

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	#35761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251258, #72251269, #72256984
maximal normal subgroup of	#72251258, #72251269, #72256984
maximal quotient of	#72308480

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6586760);