#451622

order	512
nilpotency class	3
isoclinism	isoclinic class of #424595
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16005
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

735 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)(3,14,19,34)(4,10,23,38)(5,25,28,42)(6,12)(7,30)(8,13)(9,17,33,18)(11,27,40,29)(15,43)(16,48)(20,51)(21,53)(22,37,24,41)(26,44)(31,46)(32,49)(35,47)(36,56)(39,50)(45,60,54,64)(52,63,59,61)(55,62)(57,58);(1,3)(2,9,13,33)(4,20,23,46)(5,26,28,50)(6,17)(7,18)(8,19)(10,35)(11,39)(12,14,30,34)(15,24,47,22)(16,27,49,29)(21,54)(25,48)(31,41)(32,42)(36,60,62,64)(37,51)(38,43)(40,44)(45,57)(52,56)(53,63,58,61)(55,59);(1,4,6,24,8,23,7,22)(2,10,30,37,13,38,12,41)(3,15,17,20,19,47,18,46)(5,21,29,55,28,57,27,56)(9,31,34,35,33,51,14,43)(11,36,25,58,40,62,42,53)(16,45,50,52,49,54,26,59)(32,60,39,61,48,64,44,63);(1,5)(2,11)(3,16)(4,21)(6,29)(7,27)(8,28)(9,32)(10,36)(12,42)(13,40)(14,44)(15,45)(17,50)(18,26)(19,49)(20,52)(22,56)(23,57)(24,55)(25,30)(31,60)(33,48)(34,39)(35,61)(37,58)(38,62)(41,53)(43,63)(46,59)(47,54)(51,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358818);