#6586192

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18)(19,22)(20,27)(21,25)(23,26)(24,31)(28,32)(29,30);(1,3,5,4)(2,6,7,8)(17,19,22,18)(20,28,31,25)(21,24,32,27)(23,26,30,29);(9,11,12,16)(10,13,14,15)(17,20,22,31)(18,24,19,27)(21,30,32,23)(25,26,28,29);(1,4,5,3)(2,6,7,8)(20,21)(23,30)(24,28)(25,27)(26,29)(31,32);(17,21)(18,25)(19,28)(20,30)(22,32)(23,31)(24,26)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493388);