#433563

order	512
nilpotency class	3
isoclinism	isoclinic class of #421447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2474
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15485
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

775 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340759);