#6496111

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470028
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62781	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2944 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi

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, 6403307);