#472846

order	512
nilpotency class	3
isoclinism	isoclinic class of #455690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2825
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16838
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

793 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380042);