#6354930

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354913
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1635 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262126);