#6350428

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350397
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2822 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257624);