#438499

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

936 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:

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

Polycyclic presentation:

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

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, 345695);