#498231

order	512
nilpotency class	4
isoclinism	isoclinic class of #484039
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 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)(6,8)(9,10)(11,23)(12,27,30,40)(13,19)(14,31)(15,20)(16,22)(17,25)(18,24,34,39)(21,26)(28,36,29,37)(32,33,38,35);(1,2)(3,6)(4,5)(7,8)(9,11,15,14,16,26,13,25)(10,17,19,21,22,31,20,23)(12,24,29,32,30,39,28,38)(18,27,33,36,34,40,35,37);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,24)(13,28)(14,32)(15,29)(16,30)(17,27)(19,33)(20,35)(21,36)(22,34)(23,37)(25,38)(26,39)(31,40);(2,5)(6,8)(9,13,16,15)(10,19,22,20)(11,25,26,14)(12,28,30,29)(17,21,31,23)(18,33,34,35)(24,38,39,32)(27,36,40,37)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=eh,d^2=i,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=deh,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=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, 405427);