#6347368

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	#62819	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,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, 6254564);