#6348983

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348967
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
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3001 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,25)(20,28)(21,23)(22,27)(24,29)(26,32)(30,31);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(20,29)(24,28)(25,32)(27,30);(17,19,21,26)(18,22,23,31)(20,30,29,27)(24,32,28,25);(18,23)(22,31)(24,28)(25,32);(2,7)(6,15)(8,12)(9,16)

Polycyclic presentation:

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

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, 6256179);