#6354912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354835
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	#60961	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1509 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)(3,12,16,27)(4,9,19,25)(5,10,6,20)(8,14,22,15)(13,26,30,31)(17,23,18,24)(21,28,32,29)(33,34,39,44)(35,45,47,56)(36,50,51,58)(37,43,38,42)(40,49,55,48)(41,53,57,52)(46,60,61,54)(59,63,64,62);(1,3,6,15,7,16,5,14)(2,8,10,12,11,22,20,27)(4,13,18,29,19,30,17,28)(9,21,23,26,25,32,24,31)(33,35,37,48,39,47,38,49)(34,40,42,45,44,55,43,56)(36,46,52,62,51,61,53,63)(41,54,50,59,57,60,58,64);(1,4)(2,9)(3,13)(5,17)(6,18)(7,19)(8,21)(10,23)(11,25)(12,26)(14,28)(15,29)(16,30)(20,24)(22,32)(27,31)(33,36)(34,41)(35,46)(37,52)(38,53)(39,51)(40,54)(42,50)(43,58)(44,57)(45,59)(47,61)(48,62)(49,63)(55,60)(56,64);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,17,19,18)(8,12,22,27)(9,23,25,24)(13,28,30,29)(21,26,32,31)(35,47)(40,55)(45,56)(46,61)(48,49)(54,60)(59,64)(62,63);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,19,17)(8,12,22,27)(9,24,25,23)(13,29,30,28)(21,31,32,26)(36,51)(41,57)(46,61)(50,58)(52,53)(54,60)(59,64)(62,63)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^c=eh,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, 6262108);