#509830

order	512
nilpotency class	4
isoclinism	isoclinic class of #509829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31032
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417026);