#6579488

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#120
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1396 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486684);