#496841

order	512
nilpotency class	4
isoclinism	isoclinic class of #483928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	#33519	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 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)(3,7)(4,6)(5,8)(9,10,16,19,14,20,13,27)(11,22,23,34,12,17,25,36)(15,30,32,38,29,33,28,21)(18,37,35,26,40,31,39,24);(1,3)(2,5)(4,8)(6,7)(9,11,16,23,14,12,13,25)(10,17,19,36,20,22,27,34)(15,26,32,31,29,24,28,37)(18,38,35,33,40,21,39,30);(3,8)(5,7)(9,12)(10,18)(11,14)(13,25)(15,31)(16,23)(17,33)(19,39)(20,40)(21,34)(22,30)(24,28)(26,32)(27,35)(29,37)(36,38);(2,6)(5,7)(9,13,14,16)(10,19,20,27)(11,23,12,25)(15,28,29,32)(17,34,22,36)(18,39,40,35)(21,30,38,33)(24,37,26,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ceh,d^b=di,f^b=fi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265458, #72265470, #72265492, …
maximal normal subgroup of	#72265458, #72265470, #72265492, …
maximal quotient of	#72343529, #72343562

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404037);