#6357434

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357427
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2046 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:

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

Polycyclic presentation:

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

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, 6264630);