#6489625

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470049
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	C₂²×C₄	C₂⁵
prime radical	#3006	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2834 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18)(19,27)(20,29)(21,24)(22,25)(23,26)(28,31)(30,32);(1,3,6,13)(2,7,10,16)(4,12,5,11)(8,15,9,14)(17,19)(18,20)(21,28)(22,29)(23,30)(24,27)(25,32)(26,31);(1,4,6,5)(2,8,10,9)(3,11,13,12)(7,14,16,15)(17,20)(18,19)(21,27)(22,32)(23,31)(24,28)(25,29)(26,30);(1,5,6,4)(2,9,10,8)(3,12,13,11)(7,15,16,14);(2,10)(7,16)(8,9)(14,15)(17,21,23,22)(18,24,26,25)(19,28,30,29)(20,27,31,32)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=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, 6396821);