#6353788

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

1633 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)(3,12)(4,14)(5,10)(6,13)(7,11)(8,9)(15,16)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,32)(30,31);(1,3,6,9,7,15,5,14)(2,8,10,12,11,4,13,16)(18,22)(23,31)(24,29)(25,32);(1,4)(2,9)(3,13)(5,12)(6,16)(7,8)(10,15)(11,14)(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25);(1,5,7,6)(2,10,11,13)(3,14,15,9)(4,16,8,12);(1,5,7,6)(2,10,11,13)(3,9,15,14)(4,12,8,16)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=h,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, 6260984);