#416793

order	512
nilpotency class	3
isoclinism	isoclinic class of #398772
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2439
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15232
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

621 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)(3,14,18,30)(4,9)(5,23)(6,11)(7,12)(8,17,29,40)(10,24)(13,26)(15,39,19,37)(16,28,44,55)(20,47)(21,32)(22,34)(25,35)(27,46,54,42)(31,48)(33,50)(36,52)(38,59,51,43)(41,61,45,53)(49,56)(57,64)(58,62,60,63);(1,3,7,18)(2,8)(4,19,22,15)(5,16,25,44)(6,17,26,40)(9,27)(10,28)(11,14)(12,29)(13,30)(20,45,49,41)(21,46,50,42)(23,51)(24,43,52,59)(31,53)(32,37)(33,39)(34,54)(35,38)(36,55)(47,60)(48,63,64,62)(56,58)(57,61);(1,4,6,21,7,22,26,50)(2,9,11,32,12,34,13,33)(3,15,40,46,18,19,17,42)(5,20,24,48,25,49,52,64)(8,27,30,39,29,54,14,37)(10,31,35,56,36,57,23,47)(16,41,59,63,44,45,43,62)(28,53,51,60,55,61,38,58);(1,5,6,24,7,25,26,52)(2,10,13,23,12,36,11,35)(3,16,17,43,18,44,40,59)(4,20,21,48,22,49,50,64)(8,28,30,51,29,55,14,38)(9,31,33,47,34,57,32,56)(15,41,42,62,19,45,46,63)(27,53,39,60,54,61,37,58)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=ef,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=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, 323989);