#6368269

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368046
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#18911	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh

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, 6275465);