#6353786

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

1660 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18)(19,21)(20,27)(22,25)(23,30)(24,26)(28,32)(29,31);(2,6)(7,15)(8,13)(9,16)(17,19,26,25,22,24,21,18)(20,27,30,31,28,32,29,23);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,20,22,28)(18,23,25,31)(19,27,24,32)(21,29,26,30);(20,28)(23,31)(27,32)(29,30);(17,21,22,26)(18,19,25,24)(20,29,28,30)(23,27,31,32)

Polycyclic presentation:

2^9|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, 6260982);