#452647

order	512
nilpotency class	3
isoclinism	isoclinic class of #424595
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16057
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

765 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)(3,15,19,34)(4,10,23,39)(5,25,28,41)(6,12)(7,14)(8,13)(9,18,33,30)(11,29,42,27)(16,35)(17,32)(20,31)(21,50)(22,37,24,38)(26,40)(36,51)(43,48)(44,60,64,55)(45,56)(46,58)(47,57)(49,59,63,62)(52,61)(53,54);(1,3,6,18,8,19,7,30)(2,9,14,34,13,33,12,15)(4,20,22,45,23,47,24,16)(5,26,27,46,28,48,29,17)(10,35,38,57,39,56,37,31)(11,40,25,32,42,43,41,58)(21,44,51,63,53,64,52,49)(36,60,50,59,61,55,54,62);(1,4,6,24,8,23,7,22)(2,10,14,37,13,39,12,38)(3,16,18,47,19,45,30,20)(5,21,29,51,28,53,27,52)(9,31,34,56,33,57,15,35)(11,36,41,50,42,61,25,54)(17,44,48,63,46,64,26,49)(32,55,40,62,58,60,43,59);(1,5)(2,11)(3,17)(4,21)(6,29)(7,27)(8,28)(9,32)(10,36)(12,25)(13,42)(14,41)(15,43)(16,44)(18,48)(19,46)(20,49)(22,52)(23,53)(24,51)(26,30)(31,55)(33,58)(34,40)(35,59)(37,50)(38,54)(39,61)(45,64)(47,63)(56,62)(57,60)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,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, 359843);