#498733

order	512
nilpotency class	3
isoclinism	isoclinic class of #384166
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2377
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23333
prime radical	#3050	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

454 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cefh,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405929);