#509347

order	512
nilpotency class	4
isoclinism	isoclinic class of #509344
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30593
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25447	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 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)(5,25)(6,29)(7,14)(8,36)(9,18)(10,23)(11,27)(12,35)(13,33)(16,46)(17,49)(19,53)(20,40)(22,50)(24,32)(26,47)(28,31)(30,61)(34,60)(37,42)(38,43)(39,64)(41,57)(44,54)(45,52)(48,56)(51,55)(58,63)(59,62);(1,3,8,20,7,18,33,52)(2,9,13,40,14,15,36,45)(4,22,24,58,11,16,37,48)(5,26,28,59,10,17,38,51)(6,30,35,64,34,19,54,57)(12,41,29,53,44,39,60,61)(21,46,42,63,27,50,32,56)(23,47,31,55,25,49,43,62);(1,4,7,11)(2,10,14,5)(3,16)(6,31,34,43)(8,37,33,24)(9,17)(12,42,44,32)(13,28,36,38)(15,26)(18,22)(19,55)(20,58)(21,60,27,29)(23,54,25,35)(30,62)(39,56)(40,59)(41,63)(45,51)(46,53)(47,57)(48,52)(49,64)(50,61);(1,5,7,10)(2,11,14,4)(3,17,18,26)(6,32,34,42)(8,38,33,28)(9,22,15,16)(12,43,44,31)(13,24,36,37)(19,56,30,63)(20,59,52,51)(21,35,27,54)(23,29,25,60)(39,62,41,55)(40,48,45,58)(46,57,50,64)(47,61,49,53)

Polycyclic presentation:

2^9|b^2=h,c^2=ei,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,g^c=gi,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, 416543);