#6447739

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3033 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,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, 6354935);