#6694393

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	#42529	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1447 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)(3,8,4,6)(9,10,13,15)(11,14,12,16)(18,24)(19,27)(26,32)(29,30);(1,2)(3,6)(4,8)(5,7)(9,11,13,12)(10,14,15,16)(17,18)(19,28)(20,26)(21,29)(22,24)(23,32)(25,30)(27,31);(9,12,13,11)(10,14,15,16)(17,19,22,27)(18,25,24,21)(20,30,23,29)(26,31,32,28);(1,3,5,4)(2,6,7,8)(17,20,22,23)(18,26,24,32)(19,29,27,30)(21,31,25,28);(1,4,5,3)(2,6,7,8)(17,21)(18,19)(20,31)(22,25)(23,28)(24,27)(26,29)(30,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=g,d^2=f,e^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, 6601589);