#479789

order	512
nilpotency class	3
isoclinism	isoclinic class of #479788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17248
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

597 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)(3,15,20,43)(4,21)(5,11)(6,12)(7,13)(8,14)(9,18,33,19)(10,25)(16,45,48,31)(17,44,52,59)(22,34,47,61)(23,39)(24,37)(26,36)(27,56)(28,40)(29,41)(30,42)(32,51,38,50)(35,54)(46,57,64,62)(49,58,63,60)(53,55);(1,3)(2,9)(4,22,26,47)(5,17)(6,19)(7,18)(8,20)(10,34,37,61)(11,38)(12,43)(13,15)(14,33)(16,24,48,25)(21,31,36,45)(23,49,55,63)(27,46,54,64)(28,51)(29,50)(30,52)(32,42)(35,58,56,60)(39,57,53,62)(40,59)(41,44);(1,4,7,25,8,26,6,24)(2,10,12,21,14,37,13,36)(3,16,19,22,20,48,18,47)(5,27,29,23,30,54,28,55)(9,31,15,34,33,45,43,61)(11,39,40,56,42,53,41,35)(17,49,51,46,52,63,50,64)(32,58,59,62,38,60,44,57);(1,5,8,30)(2,11,14,42)(3,17,20,52)(4,23,26,55)(6,29,7,28)(9,32,33,38)(10,35,37,56)(12,41,13,40)(15,44,43,59)(16,46,48,64)(18,51,19,50)(21,53,36,39)(22,49,47,63)(24,54,25,27)(31,57,45,62)(34,60,61,58)

Polycyclic presentation:

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

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, 386985);