#6353782

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	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1563 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,15)(12,16)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16);(1,2)(3,7)(4,14)(5,8)(6,9)(10,13)(11,16)(12,15)(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,10,15,16);(3,12)(7,15)(10,16)(11,13)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^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, 6260978);