#6351843

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
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	#62825	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2577 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)(3,13)(4,17)(5,12)(6,11)(7,10)(8,16)(9,19)(14,22)(15,25)(18,26)(20,23)(21,24)(27,30)(28,32)(29,31)(33,34,39,43)(35,44,47,54)(36,41,50,57)(37,42,38,51)(40,52,55,46)(45,60,62,59)(48,56,49,58)(53,63,64,61);(1,3)(2,8)(4,14)(5,20)(6,16)(7,15)(9,22)(10,23)(11,25)(12,13)(17,31)(18,28)(19,30)(21,29)(24,32)(26,27)(33,35)(34,40)(36,45)(37,46)(38,52)(39,47)(41,53)(42,54)(43,55)(44,51)(48,61)(49,63)(50,62)(56,59)(57,64)(58,60);(1,4,7,18)(2,9,10,24)(3,14,15,28)(5,19,6,21)(8,22,23,32)(11,17,12,26)(13,27,25,31)(16,29,20,30)(33,36,39,50)(34,41,43,57)(35,45,47,62)(37,49,38,48)(40,53,55,64)(42,58,51,56)(44,59,54,60)(46,63,52,61);(2,10)(4,18)(8,23)(11,12)(13,25)(14,28)(19,21)(29,30)(33,37,39,38)(34,42,43,51)(35,46,47,52)(36,48,50,49)(40,54,55,44)(41,56,57,58)(45,61,62,63)(53,59,64,60);(3,15)(8,23)(13,25)(14,28)(16,20)(22,32)(27,31)(29,30)(35,47)(40,55)(44,54)(45,62)(46,52)(53,64)(59,60)(61,63)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259039);