#6447746

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3023 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:

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

Polycyclic presentation:

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

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, 6354942);