#6349705

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349684
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

2570 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:

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

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256901);