#316146

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=e,c^2=fhi,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,d^c=dh

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, 223342);