#6353787

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353765
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

1692 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:

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

Polycyclic presentation:

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

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, 6260983);