#6347367

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
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3114 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)(3,13)(4,7)(5,11)(6,20)(8,12)(9,15)(10,22)(14,23)(16,29)(17,26)(18,31)(19,24)(21,25)(27,32)(28,30);(1,3)(2,9)(4,14)(5,15)(6,23)(7,16)(8,18)(10,26)(11,29)(12,28)(13,20)(17,25)(19,31)(21,27)(22,30)(24,32);(1,4,7,2)(3,14,16,9)(5,10,6,19)(8,21,25,12)(11,24,20,22)(13,30,29,32)(15,26,23,31)(17,28,18,27);(1,5,7,6)(2,10,4,19)(3,15,16,23)(8,22,25,24)(9,26,14,31)(11,21,20,12)(13,28,29,27)(17,32,18,30);(3,16)(9,14)(13,29)(15,23)(17,18)(26,31)(27,28)(30,32)

Polycyclic presentation:

2^9|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, 6254563);