#413507

order	512
nilpotency class	3
isoclinism	isoclinic class of #401030
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2541
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15148
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1013 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,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, 320703);