#6459177

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35190	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,24,27,23,19,22,28)(20,29,21,26,32,25,31,30);(2,6)(7,8)(17,19,23,18)(20,26,32,30)(21,29,31,25)(22,28,24,27);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,24,21,23,32,22,31)(18,25,27,30,19,29,28,26);(1,4,5,3)(2,8,6,7)(17,21,22,20,23,31,24,32)(18,26,28,29,19,30,27,25);(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,27,19,28)(20,31,32,21)(25,30,29,26)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366373);