#431486

order	512
nilpotency class	3
isoclinism	isoclinic class of #431485
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15439
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338682);