#459492

order	512
nilpotency class	3
isoclinism	isoclinic class of #459491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16341
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1002 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366688);