#510329

order	512
nilpotency class	4
isoclinism	isoclinic class of #510327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31495
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

433 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:

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

Polycyclic presentation:

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

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, 417525);