#6354829

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354774
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
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

1485 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh,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, 6262025);