#6581561

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1369 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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, 6488757);