#479845

order	512
nilpotency class	3
isoclinism	isoclinic class of #478580
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2789
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17276
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

499 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)(3,15,19,9)(4,20)(5,26)(6,14)(7,13)(8,12)(10,23)(11,29)(16,46,49,35)(17,39,50,60)(18,34,31,43)(21,44,48,32)(22,40)(24,37)(25,38)(27,45,57,33)(28,41)(30,42)(36,56)(47,59,53,58)(51,54)(52,61,64,63)(55,62);(1,3,7,18,8,19,6,31)(2,9,13,43,12,15,14,34)(4,21,23,16,25,48,24,49)(5,27,29,50,30,57,28,17)(10,35,38,44,37,46,20,32)(11,39,42,45,41,60,26,33)(22,53,54,52,56,47,55,64)(36,58,62,61,40,59,51,63);(1,4,6,23,8,25,7,24)(2,10,13,20,12,37,14,38)(3,16,18,21,19,49,31,48)(5,22,28,54,30,56,29,55)(9,32,34,35,15,44,43,46)(11,40,42,62,41,36,26,51)(17,47,27,52,50,53,57,64)(33,59,60,63,45,58,39,61);(1,5,6,28,8,30,7,29)(2,11,13,42,12,41,14,26)(3,17,18,27,19,50,31,57)(4,22,23,54,25,56,24,55)(9,33,34,60,15,45,43,39)(10,36,20,51,37,40,38,62)(16,47,21,52,49,53,48,64)(32,58,35,61,44,59,46,63)

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,c^2=f,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,d^c=de,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387041);