#471047

order	512
nilpotency class	3
isoclinism	isoclinic class of #460970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2759
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16741
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

704 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)(3,15,20,34)(4,21)(5,27)(6,12)(7,13)(8,14)(9,18,35,19)(10,24)(11,30)(16,44,46,32)(17,47,51,33)(22,36,52,59)(23,54)(25,39)(26,38)(28,53)(29,42)(31,41)(37,56)(40,55)(43,49,61,50)(45,60,57,62)(48,63,64,58);(1,3,7,19,8,20,6,18)(2,9,12,15,14,35,13,34)(4,22,24,46,26,52,25,16)(5,17,30,50,31,51,29,49)(10,36,21,44,39,59,38,32)(11,33,27,43,42,47,41,61)(23,48,28,57,56,64,55,45)(37,62,53,63,54,60,40,58);(1,4)(2,10)(3,16)(5,28)(6,25)(7,24)(8,26)(9,32)(11,40)(12,21)(13,38)(14,39)(15,36)(17,48)(18,52)(19,22)(20,46)(23,29)(27,37)(30,56)(31,55)(33,60)(34,59)(35,44)(41,54)(42,53)(43,58)(45,49)(47,62)(50,57)(51,64)(61,63);(1,5,6,29,8,31,7,30)(2,11,13,41,14,42,12,27)(3,17,18,49,20,51,19,50)(4,23,24,28,26,56,25,55)(9,33,34,61,35,47,15,43)(10,37,21,53,39,54,38,40)(16,45,22,48,46,57,52,64)(32,58,36,62,44,63,59,60)

Polycyclic presentation:

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

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, 378243);