#431487

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15439
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1038 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338683);