#6354285

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354266
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	#60962	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1693 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,d^b=dh,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, 6261481);