#6729677

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#2433	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1583 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=g,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,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, 6636873);