#326571

order	512
nilpotency class	3
isoclinism	isoclinic class of #252235
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2580
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,d^c=di,f^c=fhi

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, 233767);