#6445668

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2051 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352864);