#504336

order	512
nilpotency class	4
isoclinism	isoclinic class of #504336
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26529
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

435 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=fh,c^2=ef,d^2=e,e^2=i,g^2=h,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, 411532);