#472463

order	512
nilpotency class	3
isoclinism	isoclinic class of #458125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,25)(13,16)(14,20)(15,19)(18,29)(21,37)(22,31)(23,30)(24,33)(26,40)(27,36)(28,35)(32,38)(34,39);(1,3,4,7)(2,5,6,8)(10,16,20,19)(17,31,33,30)(18,25,36,35)(32,37,40,34);(9,11,15,22,14,24,13,23)(10,17,16,30,20,33,19,31)(12,26,29,39,28,38,27,21)(18,34,25,40,36,37,35,32);(2,6)(5,8)(9,12,13,27,14,28,15,29)(10,18,19,35,20,36,16,25)(11,21,22,26,24,39,23,38)(17,32,30,34,33,40,31,37)

Polycyclic presentation:

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

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, 379659);