#479791

order	512
nilpotency class	3
isoclinism	isoclinic class of #478348
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17250
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

603 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)(3,9,17,33)(4,19)(5,11,27,40)(6,14)(7,13)(8,12)(10,22)(15,34,45,61)(16,31)(18,42,28,32)(20,30,51,57)(21,52,54,49)(23,36)(24,37)(25,38)(26,43,29,41)(35,53,39,55)(44,63)(46,60)(47,59)(48,58)(50,62)(56,64);(1,3)(2,9,12,33)(4,20,24,51)(5,25,27,48)(6,28)(7,18)(8,17)(10,34)(11,38)(13,42,14,32)(15,23,45,22)(16,26,47,29)(19,30)(21,44)(31,41)(35,62,39,56)(36,61)(37,57)(40,58)(43,59)(46,54)(49,60,52,63)(50,55)(53,64);(1,4,6,22,8,24,7,23)(2,10,13,19,12,36,14,37)(3,15,18,20,17,45,28,51)(5,21,26,53,27,54,29,55)(9,30,32,34,33,57,42,61)(11,39,41,52,40,35,43,49)(16,46,48,64,47,44,25,50)(31,56,38,60,59,62,58,63);(1,5)(2,11,12,40)(3,16,17,47)(4,21)(6,26)(7,29)(8,27)(9,31)(10,35,36,39)(13,41,14,43)(15,44,45,46)(18,48,28,25)(19,49,37,52)(20,50,51,64)(22,53)(23,55)(24,54)(30,56)(32,38)(33,59)(34,60)(42,58)(57,62)(61,63)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=gi,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, 386987);