#459713

order	512
nilpotency class	3
isoclinism	isoclinic class of #459490
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2733
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16347
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

839 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=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, 366909);