#462857

order	512
nilpotency class	3
isoclinism	isoclinic class of #455393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16452
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

700 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^c=hi,h^d=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, 370053);