#6347397

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347363
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	#62835	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,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, 6254593);