#6444734

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439029
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1907 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6351930);