#6349407

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349370
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

2561 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)(4,9)(5,11)(6,10)(7,12)(8,15)(14,27)(16,22)(17,26)(18,25)(19,23)(20,24)(21,29)(28,32)(30,31)(33,34,39,44)(35,45,49,55)(36,50,51,58)(37,42,38,43)(40,48,56,47)(41,53,57,52)(46,60,61,54)(59,63,64,62);(1,3,7,17)(2,8,12,22)(4,14,20,30)(5,15,6,16)(9,21,24,32)(10,13,11,26)(18,28,19,29)(23,31,25,27)(33,35,39,49)(34,40,44,56)(36,46,51,61)(37,48,38,47)(41,54,57,60)(42,45,43,55)(50,64,58,59)(52,63,53,62);(1,4)(2,9)(3,14)(5,19)(6,18)(7,20)(8,21)(10,25)(11,23)(12,24)(13,27)(15,29)(16,28)(17,30)(22,32)(26,31)(33,36)(34,41)(35,46)(37,52)(38,53)(39,51)(40,54)(42,58)(43,50)(44,57)(45,59)(47,62)(48,63)(49,61)(55,64)(56,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,18,20,19)(8,13,22,26)(9,23,24,25)(14,28,30,29)(21,31,32,27)(36,51)(41,57)(46,61)(50,58)(52,53)(54,60)(59,64)(62,63);(1,5,7,6)(2,11,12,10)(3,16,17,15)(4,19,20,18)(8,13,22,26)(9,23,24,25)(14,28,30,29)(21,27,32,31)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,52,51,53)(40,55,56,45)(41,58,57,50)(46,62,61,63)(54,64,60,59)

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256603);