#6458562

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	#35272	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2074 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)(3,7)(4,8)(5,6)(9,10,13,15)(11,14,12,16)(17,18,22,28,24,19,23,27)(20,29,31,30,32,25,21,26);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,24,18)(20,30,32,26)(21,25,31,29)(22,28,23,27);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,22,31,24,32,23,21)(18,25,28,26,19,29,27,30);(1,4,5,3)(2,8,6,7)(17,21,23,32,24,31,22,20)(18,26,27,25,19,30,28,29);(17,22,24,23)(18,27,19,28)(20,31,32,21)(25,30,29,26)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=fg,d^2=f,e^2=i,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, 6365758);