#6354956

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1573 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=hi,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, 6262152);