#509111

order	512
nilpotency class	4
isoclinism	isoclinic class of #509109
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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,f^2=hi,g^2=i,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,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, 416307);