#6354959

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	#60946	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1643 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)(18,22)(19,26)(20,23)(25,30)(27,28)(29,32);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,18,23,30,21,25,20,22)(19,24,26,32,27,31,28,29);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,19)(18,24)(20,28)(21,27)(22,29)(23,26)(25,31)(30,32);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15);(19,27)(24,31)(26,28)(29,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,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, 6262155);