#6355081

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355044
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

1532 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)(4,15)(5,10)(6,13)(7,11)(8,9)(14,16)(17,18,22,26)(19,23,28,32)(20,25,21,30)(24,29,31,27);(1,3,6,9,7,14,5,15)(2,8,10,12,11,4,13,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(1,4)(2,9)(3,13)(5,12)(6,16)(7,8)(10,14)(11,15)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(1,5,7,6)(2,10,11,13)(3,9,14,15)(4,12,8,16)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,24,32,31);(3,14)(4,8)(9,15)(12,16)

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^b=eh,e^c=eh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262277);