#6357693

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357668
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
prime radical	#62414	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1847 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)(3,10,8,5)(4,7,12,14)(11,16,15,13)(18,24)(19,25)(21,22)(26,27)(28,29)(31,32);(1,3,5,2,6,8,10,9)(4,11,13,7,12,15,16,14)(17,18,21,24,23,27,22,26)(19,20,29,31,28,30,25,32);(1,4)(2,7)(3,11)(5,13)(6,12)(8,15)(9,14)(10,16)(17,19,21,29,23,28,22,25)(18,20,24,31,27,30,26,32);(1,4)(2,7)(3,11)(5,13)(6,12)(8,15)(9,14)(10,16)(17,20)(18,25)(19,24)(21,31)(22,32)(23,30)(26,28)(27,29);(4,12)(7,14)(11,15)(13,16)(19,28)(20,30)(25,29)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264889);