#441631

order	512
nilpotency class	3
isoclinism	isoclinic class of #427841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2778
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15688
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

868 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)(3,15,19,40)(4,20,23,10)(5,25,28,38)(6,12)(7,14)(8,13)(9,18,33,30)(11,29,39,27)(16,41)(17,32)(21,51)(22,36,24,35)(26,37)(31,44)(34,55)(42,48)(43,63,64,57)(45,58)(46,59)(47,49)(50,53)(52,62,56,60)(54,61);(1,3,8,30,7,19,6,18)(2,9,12,15,14,33,13,40)(4,16,24,47,23,45,22,44)(5,26,29,17,28,48,27,46)(10,31,35,58,20,49,36,41)(11,37,38,59,39,42,25,32)(21,52,55,43,54,56,53,64)(34,60,51,63,50,62,61,57);(1,4,8,22,7,23,6,24)(2,10,12,36,14,20,13,35)(3,16,18,47,19,45,30,44)(5,21,27,55,28,54,29,53)(9,31,40,58,33,49,15,41)(11,34,25,51,39,50,38,61)(17,43,48,52,46,64,26,56)(32,57,37,62,59,63,42,60);(1,5)(2,11)(3,17)(4,21)(6,29)(7,28)(8,27)(9,32)(10,34)(12,25)(13,38)(14,39)(15,42)(16,43)(18,48)(19,46)(20,50)(22,55)(23,54)(24,53)(26,30)(31,57)(33,59)(35,61)(36,51)(37,40)(41,60)(44,56)(45,64)(47,52)(49,63)(58,62)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,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, 348827);