#498539

order	512
nilpotency class	4
isoclinism	isoclinic class of #484139
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	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

454 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309600, #72309603, #72309613, …
maximal normal subgroup of	#72309600, #72309603, #72309613, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405735);