#6356516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356405
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	#537	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1444 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263712);