#315703

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
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

644 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)(13,15)(14,19)(16,18)(20,23)(22,24)(25,26)(27,35)(28,34)(29,31)(30,33)(32,36)(37,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,11,18,24,13,20,12,19)(10,14,21,23,15,22,16,17)(26,31)(32,39)(33,34)(35,40);(3,7)(5,8)(10,15)(14,22)(16,21)(17,23)(25,27,30,36,29,37,28,38)(26,32,34,35,31,39,33,40);(9,11,18,24,13,20,12,19)(10,14,21,23,15,22,16,17)(25,28,29,30)(26,33,31,34)(27,36,37,38)(32,35,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,c^2=fh,d^2=eh,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^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222899);