#435708

order	512
nilpotency class	3
isoclinism	isoclinic class of #420517
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15534
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

722 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,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, 342904);