#510330

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	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

441 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=i,d^2=ehi,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, 417526);