#504343

order	512
nilpotency class	4
isoclinism	isoclinic class of #500737
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3325
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#26533
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

429 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=fh,c^2=efh,d^2=e,e^2=i,f^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,d^c=di,f^c=fi,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, 411539);