#499518

order	512
nilpotency class	4
isoclinism	isoclinic class of #13523
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2092
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2092
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

717 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=i,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320511, #72320647, #72320688
maximal normal subgroup of	#72320511, #72320647, #72320688
maximal quotient of	#72322889

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406714);