#509116

order	512
nilpotency class	4
isoclinism	isoclinic class of #509113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30400
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28180	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

644 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)(4,21)(5,24)(6,27)(7,34)(8,13)(9,18)(10,22)(11,25)(12,32)(14,31)(16,46)(17,49)(19,53)(20,37)(23,43)(26,42)(28,59)(29,40)(30,38)(33,58)(35,50)(36,60)(39,47)(41,57)(44,54)(45,52)(48,62)(51,64)(55,61)(56,63);(1,3,7,20,8,18,31,52)(2,9,14,37,13,15,34,45)(4,17,23,51,5,16,26,48)(6,28,32,60,33,19,54,57)(10,35,38,61,11,39,40,63)(12,41,27,53,44,36,58,59)(21,46,42,64,24,49,43,62)(22,47,29,55,25,50,30,56);(1,4,8,5)(2,10,13,11)(3,16,18,17)(6,29,33,30)(7,26,31,23)(9,35,15,39)(12,42,44,43)(14,40,34,38)(19,55,28,56)(20,51,52,48)(21,58,24,27)(22,54,25,32)(36,62,41,64)(37,63,45,61)(46,59,49,53)(47,60,50,57);(1,5,8,4)(2,11,13,10)(3,17,18,16)(6,30,33,29)(7,23,31,26)(9,35,15,39)(12,43,44,42)(14,38,34,40)(19,56,28,55)(20,48,52,51)(21,27,24,58)(22,32,25,54)(36,62,41,64)(37,63,45,61)(46,59,49,53)(47,57,50,60)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fh,h^c=hi,f^d=fh,h^d=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, 416312);