#497144

order	512
nilpotency class	4
isoclinism	isoclinic class of #483993
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	#33717	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

701 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=efh,d^2=ei,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264749, #72264781, #72265360, …
maximal normal subgroup of	#72264749, #72264781, #72265360, …
maximal quotient of	#72343535, #72343565

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404340);