#6489190

order	512
nilpotency class	3
isoclinism	isoclinic class of #6489190
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#35237
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1734 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396386);