#6354176

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354139
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

1666 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)(3,11)(4,10)(5,7)(6,9)(8,12)(13,16)(14,15)(17,18,22,24)(19,21,26,31)(20,27,29,25)(23,28,32,30);(2,7)(8,15)(9,10)(11,16)(17,19,21,24,22,26,31,18)(20,25,28,32,29,27,30,23);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,11,15,16)(17,21,22,31)(18,19,24,26)(20,28,29,30)(23,25,32,27);(20,29)(23,32)(25,27)(28,30)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^c=eh,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, 6261372);