#509121

order	512
nilpotency class	4
isoclinism	isoclinic class of #509119
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	#24297	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

648 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fh,h^c=hi,e^d=ei,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, 416317);