#6349367

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349298
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

2569 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2,7,11)(3,12,16,25)(4,9,17,23)(5,10,6,20)(8,14,22,15)(13,26,27,32)(18,24,19,30)(21,28,31,29)(33,34,39,44)(35,45,48,55)(36,50,53,57)(37,43,38,42)(40,47,56,49)(41,51,58,52)(46,60,62,54)(59,63,64,61);(1,3,7,16)(2,8,11,22)(4,13,17,27)(5,15,6,14)(9,21,23,31)(10,25,20,12)(18,29,19,28)(24,32,30,26)(33,35,39,48)(34,40,44,56)(36,46,53,62)(37,47,38,49)(41,54,58,60)(42,55,43,45)(50,64,57,59)(51,61,52,63);(1,4,7,17)(2,9,11,23)(3,13,16,27)(5,18,6,19)(8,21,22,31)(10,24,20,30)(12,26,25,32)(14,28,15,29)(33,36,39,53)(34,41,44,58)(35,46,48,62)(37,52,38,51)(40,54,56,60)(42,50,43,57)(45,59,55,64)(47,63,49,61);(4,17)(9,23)(13,27)(18,19)(21,31)(24,30)(26,32)(28,29)(33,37,39,38)(34,42,44,43)(35,47,48,49)(36,51,53,52)(40,55,56,45)(41,57,58,50)(46,61,62,63)(54,59,60,64);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,17,19)(8,12,22,25)(9,24,23,30)(13,28,27,29)(21,26,31,32)(35,48)(40,56)(45,55)(46,62)(47,49)(54,60)(59,64)(61,63)

Polycyclic presentation:

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

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, 6256563);