#6445671

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	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=g,e^2=i,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, 6352867);