#509106

order	512
nilpotency class	4
isoclinism	isoclinic class of #509105
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30396
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#444 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

649 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)(3,16)(4,21)(5,15)(6,29)(7,12)(8,14)(9,18)(10,19)(11,41)(13,33)(17,52)(20,47)(22,39)(23,44)(24,43)(25,58)(26,49)(27,38)(28,35)(30,37)(31,50)(32,45)(34,48)(36,53)(40,61)(42,59)(46,60)(51,56)(54,57)(55,63)(62,64);(1,3)(2,9)(4,22)(5,27)(6,30)(7,18)(8,20)(10,13)(11,42)(12,47)(14,38)(15,16)(17,23)(19,34)(21,57)(24,59)(25,54)(26,55)(28,31)(29,40)(32,53)(33,61)(35,37)(36,43)(39,44)(41,64)(45,51)(46,62)(48,50)(49,52)(56,60)(58,63);(1,3)(2,10)(4,17)(5,18)(6,31)(7,27)(8,20)(9,35)(11,39)(12,40)(13,38)(14,37)(15,50)(16,29)(19,33)(21,36)(22,23)(24,56)(25,55)(26,54)(28,30)(32,53)(34,61)(41,64)(42,58)(43,52)(44,51)(45,63)(46,57)(47,48)(49,62)(59,60);(1,4,8,26)(2,11,14,45)(3,17,20,54)(5,25,7,23)(6,32,34,41)(9,36,38,62)(10,39,37,63)(12,46,15,43)(13,49,35,21)(16,51,47,42)(18,55,27,22)(19,56,30,59)(24,28,60,33)(29,44,48,58)(31,53,61,64)(40,57,50,52)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416302);