#498233

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	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

553 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,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.

Relationships

maximal subgroup of	#72264905, #72264915, #72264937, …
maximal normal subgroup of	#72264905, #72264915, #72264937, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405429);