#316742

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

656 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)(3,7)(4,6)(5,8)(9,10,15,19)(11,16,12,21)(13,18,14,23)(17,20,24,22)(25,26,31,34)(27,32,28,38)(29,39,30,35)(33,36,40,37);(1,3,4,8)(2,5,6,7)(9,11,15,12)(10,16,19,21)(13,22,14,20)(17,23,24,18)(25,27,29,36,31,28,30,37)(26,32,39,40,34,38,35,33);(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18)(25,28,31,27)(26,33,34,40)(29,36,30,37)(32,39,38,35);(2,6)(5,7)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)(26,34)(32,38)(33,40)(35,39)

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,d^b=dhi,f^c=fh,g^c=gi

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, 223938);