#6579516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,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, 6486712);