#501735

order	512
nilpotency class	4
isoclinism	isoclinic class of #501734
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24812
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

497 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=e,d^2=ei,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=gh,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, 408931);